Document:

Exhibit 10.5

FORM OF PRIVATE PLACEMENT WARRANTS
PURCHASE AGREEMENT

THIS PRIVATE PLACEMENT WARRANTS PURCHASE
AGREEMENT, dated as of [___], 2022 (as it may from time to time be amended, this “Agreement”), is entered into by and
between Apollo Strategic Growth Capital III, a Cayman Islands exempted company, incorporated with limited liability (the “Company”),
and APSG Sponsor III, L.P., a Cayman Islands exempted limited partnership (the “Purchaser”).

WHEREAS, the Company intends to consummate
an initial public offering of the Company’s units (the “Public Offering”), each unit consisting of one Class
A ordinary share, par value $0.0001 per share, of the Company (each, a “Share”) and a fraction of one redeemable warrant
as set forth in the Company’s registration statement on Form S-1, filed with the Securities and Exchange Commission (the
 “SEC”), File Number 333-254835 (the “Registration Statement”), under the Securities Act of 1933, as amended
(the “Securities Act”). Each whole warrant entitles the holder to purchase one Share at an exercise price of $11.50
per share. The Purchaser has agreed to purchase an aggregate of 6,107,500 warrants (or up to 6,807,500 warrants if the underwriters’
over-allotment option in connection with the Public Offering is exercised in full) (the “Private Placement Warrants”),
each whole Private Placement Warrant entitling the holder to purchase one Share at an exercise price of $11.50 per share.

NOW THEREFORE, in consideration of the
mutual promises contained in this Agreement and other good and valuable consideration, the receipt and sufficiency of which are
hereby acknowledged, the parties to this Agreement hereby, intending legally to be bound, agree as follows:

AGREEMENT

		Section 1.	Authorization, Purchase and Sale; Terms of the Private Placement Warrants.

A.               
Authorization of the Private Placement Warrants. The Company has duly authorized the issuance and sale of the Private
Placement Warrants to the Purchaser.

B.                
Purchase and Sale of the Private Placement Warrants.

(i)                
On the date of the closing of the Public Offering or on such earlier time and date as may be mutually agreed by the Purchaser
and the Company (the “Initial Closing Date”), the Company shall issue and sell to the Purchaser, and the Purchaser
shall purchase from the Company, 6,107,500 Private Placement Warrants at a price of $1.50 per warrant for an aggregate purchase
price of $9,161,250 (the “Purchase Price”). The Purchaser shall pay the Purchase Price by wire transfer of immediately
available funds to the Company in accordance with the Company’s wiring instructions. On the Initial Closing Date, upon the
payment by the Purchaser of the Purchase Price, the Company shall either, at its option, shall deliver a certificate evidencing
the Private Placement Warrants purchased on such date duly registered in the Purchaser’s name to the Purchaser, or effect
such delivery in book-entry form.

    	  

    	 

    

(ii)              
 On the date of the closing of the underwriters’ over-allotment option in connection with the Public Offering or on
such earlier time and date as may be mutually agreed by the Purchaser and the Company (each such date, an “Over-allotment
Closing Date”, and each Over-allotment Closing Date (if any) and the Initial Closing Date being sometimes referred to herein
as a “Closing Date”), the Company shall issue and sell to the Purchaser, and the Purchaser shall purchase from the
Company, up to 700,000 Private Placement Warrants, in the same proportion as the amount of the underwriters’ over-allotment
option that is exercised, at a price of $1.50 per warrant for an aggregate purchase price of up to $1,050,000 (if the underwriters’
over-allotment option in connection with the Public Offering is exercised in full) (the “Over-allotment Purchase Price”).
The Purchaser shall pay the Over-allotment Purchase Price by wire transfer of immediately available funds to the Company in accordance
with the Company’s wiring instructions. On the Over-allotment Closing Date, upon the payment by the Purchaser of the Over-allotment
Purchase Price, the Company shall either, at its option, deliver a certificate evidencing the Private Placement Warrants duly registered
in the Purchaser’s name to the Purchaser, or effect such delivery in book-entry form.

C.                
Terms of the Private Placement Warrants.

(i)                 Each
Private Placement Warrant shall have the terms set forth in a Private Warrant Agreement to be entered into by the Company and a
warrant agent, in connection with the Public Offering (the “Private Warrant Agreement”).

(ii)              
At or prior to the time of the closing of the Public Offering, the Company and the Purchaser shall enter into a registration
rights agreement (the “Registration Rights Agreement”) pursuant to which the Company will grant certain registration
rights to the Purchaser relating to the Private Placement Warrants and the Shares underlying the Private Placement Warrants.

		Section 2.	Representations and Warranties of the Company. As a material inducement to the Purchaser to enter into this Agreement and
purchase the Private Placement Warrants, the Company hereby represents and warrants to the Purchaser (which representations and
warranties shall survive each Closing Date) that:

A.               
Organization and Corporate Power. The Company is an exempted company, duly incorporated, validly existing and in
good standing under the laws of the Cayman Islands and is qualified to do business in every jurisdiction in which the failure to
so qualify would reasonably be expected to have a material adverse effect on the financial condition, operating results or assets
of the Company. The Company possesses all requisite corporate power and authority necessary to carry out the transactions contemplated
by this Agreement and the Private Warrant Agreement.

B.                
Authorization; No Breach.

(i)                 The
execution, delivery and performance of this Agreement and the Private Placement Warrants have been duly authorized by the Company as
of the Closing Date. This Agreement constitutes a valid and binding obligation of the Company, enforceable in accordance with its
terms. Upon issuance in accordance with, and payment pursuant to, the terms of the Private Warrant Agreement and this Agreement, the
Private Placement Warrants will constitute valid and binding obligations of the Company, enforceable in accordance with their terms
as of the Closing Date.

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(ii)              
The execution and delivery by the Company of this Agreement and the Private Placement Warrants, the issuance and sale of
the Private Placement Warrants, the issuance of the Shares upon exercise of the Private Placement Warrants and the fulfillment
of, and compliance with, the respective terms hereof and thereof by the Company, do not and will not as of the Closing Date (a)
conflict with or result in a breach of the terms, conditions or provisions of, (b) constitute a default under, (c) result in the
creation of any lien, security interest, charge or encumbrance upon the Company’s share capital or assets under, (d) result
in a violation of, or (e) require any authorization, consent, approval, exemption or other action by or notice or declaration to,
or filing with, any court or administrative or governmental body or agency pursuant to the amended and restated memorandum and
articles of association of the Company (in effect on the date hereof or as may be amended prior to completion of the contemplated
Public Offering), or any material law, statute, rule or regulation to which the Company is subject, or any agreement, order, judgment
or decree to which the Company is subject, except for any filings required after the date hereof under federal or state securities
laws.

C.                 Title
to Securities. Upon issuance in accordance with, and payment pursuant to, the terms hereof and the Private Warrant Agreement,
the Private Placement Warrants will be duly and validly issued and the Shares issuable upon exercise of the Private Placement
Warrants will be duly and validly issued, fully paid and non-assessable. On the date of issuance of the Private Placement Warrants,
the Shares issuable upon exercise of the Private Placement Warrants shall have been reserved for issuance. Upon issuance in
accordance with, and payment pursuant to, the terms hereof and the Private Warrant Agreement, the Purchaser will have good title to
the Private Placement Warrants and the Shares issuable upon exercise of such Private Placement Warrants, free and clear of all
liens, claims and encumbrances of any kind, other than (i) transfer restrictions hereunder and under the other agreements
contemplated hereby, (ii) transfer restrictions under federal and state securities laws, and (iii) liens, claims or encumbrances
imposed due to the actions of the Purchaser.

D.               
Governmental Consents. No permit, consent, approval or authorization of, or declaration to or filing with, any governmental
authority is required in connection with the execution, delivery and performance by the Company of this Agreement or the consummation
by the Company of any other transactions contemplated hereby.

E.                
Regulation D Qualification. Neither the Company nor, to its knowledge, any of its affiliates, directors or beneficial
stockholders of 20% or more of its outstanding securities, has experienced a disqualifying event as enumerated pursuant to Rule
506(d) of Regulation D under the Securities Act.

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		Section 3.	Representations and Warranties of the Purchaser.
As a material inducement to the Company to enter into this Agreement and issue and sell the Private Placement Warrants to the Purchaser,
the Purchaser hereby represents and warrants to the Company (which representations and warranties shall survive each Closing Date)
that:

A.               
Organization and Requisite Authority. The Purchaser possesses all requisite power and authority necessary to carry
out the transactions contemplated by this Agreement.

B.                
Authorization; No Breach.

(i)                
This Agreement constitutes a valid and binding obligation of the Purchaser, enforceable in accordance with its terms, subject
to bankruptcy, insolvency, fraudulent conveyance, reorganization, moratorium and other laws of general applicability relating to
or affecting creditors’ rights and to general equitable principles (whether considered in a proceeding in equity or law).

(ii)              
The execution and delivery by the Purchaser of this Agreement and the fulfillment of and compliance with the terms hereof
by the Purchaser does not and shall not as of each Closing Date conflict with or result in a breach by the Purchaser of the terms,
conditions or provisions of any agreement, instrument, order, judgment or decree to which the Purchaser is subject.

C.                
Investment Representations.

(i)                
The Purchaser is acquiring the Private Placement Warrants and, upon exercise of the Private Placement Warrants, the Shares
issuable upon such exercise (collectively, the “Securities”), for the Purchaser’s own account, for investment
purposes only and not with a view towards, or for resale in connection with, any public sale or distribution thereof.

(ii)              
The Purchaser is an “accredited investor” as such term is defined in Rule 501(a)(3) of Regulation D under the
Securities Act, and the Purchaser has not experienced a disqualifying event as enumerated pursuant to Rule 506(d) of Regulation
D under the Securities Act.

(iii)            
The Purchaser understands that the Securities are being offered and will be sold to it in reliance on specific exemptions
from the registration requirements of the United States federal and state securities laws and that the Company is relying upon
the truth and accuracy of, and the Purchaser’s compliance with, the representations and warranties of the Purchaser set forth
herein in order to determine the availability of such exemptions and the eligibility of the Purchaser to acquire such Securities.

(iv)            
The Purchaser did not enter into this Agreement as a result of any general solicitation or general advertising within the
meaning of Rule 502(c) under the Securities Act.

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(v)              
 The Purchaser has been furnished with all materials relating to the business, finances and operations of the Company and
materials relating to the offer and sale of the Securities which have been requested by the Purchaser. The Purchaser has been afforded
the opportunity to ask questions of the executive officers and directors of the Company. The Purchaser understands that its investment
in the Securities involves a high degree of risk and it has sought such accounting, legal and tax advice as it has considered necessary
to make an informed investment decision with respect to the acquisition of the Securities.

(vi)            
The Purchaser understands that no United States federal or state agency or any other government or governmental agency has
passed on or made any recommendation or endorsement of the Securities or the fairness or suitability of the investment in the Securities
by the Purchaser nor have such authorities passed upon or endorsed the merits of the offering of the Securities.

(vii)          
The Purchaser understands that: (a) the Securities have not been and are not being registered under the Securities Act or
any state securities laws, and may not be offered for sale, sold, assigned or transferred unless (1) subsequently registered thereunder
or (2) sold in reliance on an exemption therefrom; and (b) except as specifically set forth in the Registration Rights Agreement,
neither the Company nor any other person is under any obligation to register the Securities under the Securities Act or any state
securities laws or to comply with the terms and conditions of any exemption thereunder.

(viii)        
The Purchaser has such knowledge and experience in financial and business matters, understands the high degree of risk associated
with investments in the securities of companies in the development stage such as the Company, is capable of evaluating the merits
and risks of an investment in the Securities and is able to bear the economic risk of an investment in the Securities in the amount
contemplated hereunder for an indefinite period of time. The Purchaser has adequate means of providing for its current financial
needs and contingencies and will have no current or anticipated future needs for liquidity which would be jeopardized by the investment
in the Securities. The Purchaser can afford a complete loss of its investment in the Securities.

		Section 4.	Conditions of the Purchaser’s Obligations. The obligations of the Purchaser to purchase and pay for the Private Placement
Warrants are subject to the fulfillment, on or before each Closing Date, of each of the following conditions:

A.               
Representations and Warranties. The representations and warranties of the Company contained in Section 2 shall be
true and correct at and as of such Closing Date as though then made.

B.                
Performance. The Company shall have performed and complied with all agreements, obligations and conditions contained
in this Agreement that are required to be performed or complied with by it on or before such Closing Date.

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C.                  Corporate
Consents. The Company shall have obtained the consent of its Board of Directors authorizing the execution, delivery and
performance of this Agreement and the Private Warrant Agreement and the issuance and sale of the Private Placement Warrants
hereunder.

D.               
No Injunction. No litigation, statute, rule, regulation, executive order, decree, ruling or injunction shall have
been enacted, entered, promulgated or endorsed by or in any court or governmental authority of competent jurisdiction or any self-regulatory
organization having authority over the matters contemplated hereby, which prohibits the consummation of any of the transactions
contemplated by this Agreement or the Private Warrant Agreement.

E.                 Private
Warrant Agreement and Registration Rights Agreement. The Company shall have entered into the Private Warrant Agreement and the
Registration Rights Agreement, each on terms satisfactory to the Purchaser.

		Section 5.	Conditions of the Company’s Obligations. The obligations of the Company to the Purchaser under this Agreement are
subject to the fulfillment, on or before each Closing Date, of each of the following conditions:

A.               
Representations and Warranties. The representations and warranties of the Purchaser contained in Section 3 shall
be true and correct at and as of such Closing Date as though then made.

B.                
Performance. The Purchaser shall have performed and complied with all agreements, obligations and conditions contained
in this Agreement that are required to be performed or complied with by the Purchaser on or before such Closing Date.

C.                
No Injunction. No litigation, statute, rule, regulation, executive order, decree, ruling or injunction shall have
been enacted, entered, promulgated or endorsed by or in any court or governmental authority of competent jurisdiction or any self-regulatory
organization having authority over the matters contemplated hereby, which prohibits the consummation of any of the transactions
contemplated by this Agreement or the Private Warrant Agreement.

D.                Private
Warrant Agreement. The Company shall have entered into the Private Warrant Agreement on terms satisfactory to the Company.

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		Section 6.	 Termination. This Agreement may be terminated
at any time after December 31, 2021 upon the election by either the Company or the Purchaser upon written notice to the other party
if the closing of the Public Offering does not occur prior to such date.

		Section 7.	Survival of Representations and Warranties. All of the representations and warranties contained herein shall survive each
Closing Date.

		Section 8.	Definitions. Terms used but not otherwise defined in this Agreement shall have the meaning assigned to such terms in the
Registration Statement.

		Section 9.	Miscellaneous.

A.               
Successors and Assigns. Except as otherwise expressly provided herein, all covenants and agreements contained in
this Agreement by or on behalf of any of the parties hereto shall bind and inure to the benefit of the respective successors of
the parties hereto whether so expressed or not. Notwithstanding the foregoing or anything to the contrary herein, the parties may
not assign this Agreement without the prior written consent of the other party hereto, other than assignments by the Purchaser
to affiliates thereof (including, without limitation one or more of its partners or members).

B.                
Severability. Whenever possible, each provision of this Agreement shall be interpreted in such manner as to be effective
and valid under applicable law, but if any provision of this Agreement is held to be prohibited by or invalid under applicable
law, such provision shall be ineffective only to the extent of such prohibition or invalidity, without invalidating the remainder
of this Agreement.

C.                
Counterparts. This Agreement may be executed simultaneously in two or more counterparts, none of which need contain
the signatures of more than one party, but all such counterparts taken together shall constitute one and the same agreement. In
the event that any signature is delivered by facsimile transmission or by e-mail delivery of a “pdf” format data file,
such signature shall create a valid and binding obligation of the party executing (or on whose behalf such signature is executed)
with the same force and effect as if such facsimile or “.pdf” signature page were an original thereof.

D.               
Descriptive Headings; Interpretation. The descriptive headings of this Agreement are inserted for convenience only
and do not constitute a substantive part of this Agreement. The use of the word “including” in this Agreement shall
be by way of example rather than by limitation.

E.                
Governing Law. This Agreement shall be deemed to be a contract made under the laws of the State of New York and for
all purposes shall be construed in accordance with the internal laws of the State of New York.

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F.                 
 Amendments. This Agreement may not be amended, modified or waived as to any particular provision, except by a written
instrument executed by all parties hereto.

[Signature page follows]

 

    	8 

    	 

    

IN WITNESS WHEREOF, the parties
hereto have executed this Agreement to be effective as of the date first set forth above.

	 	COMPANY:
	 	 	 	 
	 	APOLLO STRATEGIC GROWTH CAPITAL III
	 	 	 	 
	 	By:	
	 	Name:	James Crossen
	 	Title:	Chief Financial Officer, Chief Accounting Officer and Secretary

	 	PURCHASER:
	 	 	 	 
	 	APSG SPONSOR III, L.P.
	 	 	 	 
	 	By: AP Caps II Holdings GP, LLC, its general partner
	 	 
	 	By: Apollo Principal Holdings
III, L.P., its managing member
	 	 
	 	By; Apollo Principal Holdings
III GP, Ltd., its general partner
	 	 
	 	By:	 
	 	Name:

                                                          Title:
	

 

 

[Signature Page to Private Placement Warrant Purchase Agreement]Exhibit 10.7

APOLLO STRATEGIC GROWTH CAPITAL
III

9 West 57th Street, 43rd Floor

New York, NY 10019

[____], 2022

APSG Sponsor III, L.P.

9 West 57th Street, 43rd Floor

New York, NY 10019

Re:       Administrative Services Agreement

 

Gentlemen:

This letter agreement by and between Apollo
Strategic Growth Capital III (the “Company”) and APSG Sponsor III, L.P. (“Sponsor”), dated as of the date
hereof, will confirm our agreement that, commencing on the date that securities of the Company are first listed on the New York
Stock Exchange (the “Listing Date”) and continuing until the earlier of the consummation by the Company of an initial
business combination or the Company’s liquidation (in each case as described in the Company’s Registration Statement
on Form S-1 (File No. 333-254835), as amended, filed with the Securities and Exchange Commission) (such earlier date hereinafter
referred to as the “Termination Date”):

(i) Sponsor shall make available, or cause
to be made available, to the Company, at 9 West 57th Street, 43rd Floor, New York, NY, 10019 (or any successor
location of Sponsor), certain office space, utilities, secretarial support and administrative services as may be reasonably requested
by the Company. In exchange therefor, the Company shall pay Sponsor the sum of $16,667 per month, for up to 24 months, commencing
on the Listing Date and continuing monthly thereafter until the Termination Date; and

(ii) Sponsor hereby irrevocably waives
any and all right, title, interest, causes of action and claims of any kind or nature as a result of, or arising out of, this letter
agreement (each, a “Claim”) in or to, and any and all right to seek payment of any amounts due to it out of, the trust
account established for the benefit of the public shareholders of the Company and into which substantially all of the proceeds
of the Company’s initial public offering will be deposited (the “Trust Account”), and hereby irrevocably waives
any Claim it may have in the future as a result of, or arising out of, this letter agreement, which Claim would reduce, encumber
or otherwise adversely affect the Trust Account or any monies or other assets in the Trust Account, and further agrees not to seek
recourse, reimbursement, payment or satisfaction of any Claim against the Trust Account or any monies or other assets in the Trust
Account for any reason whatsoever.

This letter agreement constitutes the entire
agreement and understanding of the parties hereto in respect of its subject matter and supersedes all prior understandings, agreements,
or representations by or among the parties hereto, written or oral, to the extent they relate in any way to the subject
matter hereof or the transactions contemplated hereby.

    	 

    	 

    

This letter agreement may not be amended,
modified or waived as to any particular provision, except by a written instrument executed by the parties hereto.

No party hereto may assign either this
letter agreement or any of its rights, interests, or obligations hereunder without the prior written approval of the other party;
provided that the Sponsor may assign this letter agreement or any of its rights, interests or obligations hereunder to an affiliate
without the prior written approval of the Company. Any purported assignment in violation of this paragraph shall be void and ineffectual
and shall not operate to transfer or assign any interest or title to the purported assignee.

This letter agreement, the entire relationship
of the parties hereto, and any litigation between the parties (whether grounded in contract, tort, statute, law or equity) shall
be governed by, construed in accordance with, and interpreted pursuant to the laws of the State of New York, without giving effect
to its choice of laws principles.

This letter agreement may be executed in
one or more counterparts, each of which shall for all purposes be deemed to be an original but all of which together shall constitute
one and the same letter agreement.

 

[Signature page follows]

    	 

    	 

    

 

	 	Very truly yours,
	 	 
	 	APOLLO STRATEGIC GROWTH CAPITAL III
	 	 
	 	By:	
	 	 	Name:	James Crossen
	 	 	Title:	Chief Financial Officer and Chief Accounting Officer

AGREED TO AND ACCEPTED BY:

APSG SPONSOR III, L.P.

 

	By:	 	 	 
	 	Name:	 	 
	 	Title:

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