From the following list of programs, select ten that would be most essential as 'base programs' for a fresh install of Windows:

OpenGL Wrapper
Mesa3D (get version)

3D Modelling Software
MilkShape 3D 1.8.4 (FV)
Meshx 2.4 (FV)
Terragen Classic 0.9.43 (FV)

Anti-Malware and Anti-Viruses
Clamwin 0.99.1 - Be sure to disable KernelEx if using this program! (CS)
Clam Sentinel
Avast 4.8 (CS) - Still receives malware definition updates as of May 2016
SpywareBlaster 4.6 (FV)

Audio Editor
Audacity 2.0.0 (FV)

Benchmark Software (Hard Drive)
Roadkil's Disk Speed 2.0 (FV)

CD Burning Software
Imgburn (CS)
PowerISO (FV) - 2017 builds do not work; need to find older version.
 
Cloud Storage and FTP

Codecs
ACE Mega CoDecS Pack 6.03 (FV) - Works on GOM Player, but not Zoom Player.
ffdshow rev2322 (FV)
DirectVobSub 2.39 (FV)
L3codec.ax 1.5.0.50 (FV)

**If using codecs, remember to change player settings to allow decoders. Avoid installing multiple codec packs to prevent issues.

Coding Languages
AutoIt 3.2.12.1 (FV)
Python 2.7 - Unofficial build supports Windows 95! (FV)

Customization
Talisman Desktop 3.1 (FV)

Document Viewer (PDFs)
FoxitReader 3.0 - Unofficial backport...does not properly work on Windows 9x (memory issues), but works on Windows NT 4.0 fine. (FV)
Microsoft Reader 2.1.1 (FV)
Adobe Reader 6 (FV)
SumatraPDF 0.7 (FV)
Ghostscript 8.3 - Use final .exe version for Windows 98. Allows viewing of PDFs in Irfanview. (FV)

Digital Art/Illustration/Art Software
Paint.NET 2.72 (FV)
PhotoFiltre 7.2.1 (May 8, 2015) (FV?)
Image Folio 4.30 (FV)
Artweaver 0.5 (FV)
Manga Studio 3 ($$$,  FV)

Download Accelerator
Star Downloader 1.52 - NOT compatible with Retrozilla!! (FV)
 
Driver (Hardware) Backup
Driver Genius Professional Edition 2007 (FV)

Encryption Software
TrueCrypt 1.0 (FV)

File Achieve Software
7-Zip 9.20 (FV) - However, Windows 98 displays an unspecified error with this version when drag-and-dropping files. Windows ME is free of this error for version 9.20. 7-Zip 9.22 alpha, and 9.30 works as well.
WinZip 14.5 (FV)
WinRAR 3.93 (FV)
PeaZip 5.5.1 (FV) - Supports NT 4.0 as well
Freebyte Zip 2.3.1 (2011) (FV)
Make Zip 1.29 (FV)

File Recovery
TestDisk 7.2 (CS)

Hardware Identifier

IDE (Integrated Development Environment)
Qt 4.4.3 - Assistant cannot load help files due to bug. (FV)
Qt 4.5.3 - WebKit, OpenGL, and Phonon no longer work. (US)

Image Viewers
FastStone Image Viewer 5.3 (October 24, 2014) (FV)
FastStone Image Viewer 6.4 (September 15, 2017) (US)
Irfanview 4.44 (FV)
Xnview 2.13 (FV)
Imagine 1.1.0 ANSI Edition (FV)
CIV (Common Image Viewer) (FV)
Cornice 0.6.1 (FV)

Internet Tools
OpenSSL 1.0.2t - Blackwingcat's unofficial builds support Windows 95!

Isolation Software

Math Software
GeoGebra 5.0.0  (FV) (May 12, 2015) -
Delete the C:\Program Files\GeoGebra 5.0\jre directory and replace it with the one from the OpenJDK

Media Converter

Media Players
TCPMP Player (FV)
VLC 0.8.6i (FV)
Zoom Player 14.5  (FV) - Requires GDIPLUS.dll; program may not detect codecs properly.
GOM Player 2.1.33.5071 (2.1.28.5039???) (FV) - Be sure to install codecs otherwise MP4 cannot play.
BSPlayer 2.x (???) - Have yet to find a working version.
Quickview 2.61 ($$$, CS) - MS-DOS program with MP4 support!

MP3/Music Player
XMPlay (CS)
The Core Media Player 4.11 build 452 (~$, FV)
Trout Audio Player 1.0.6 build 71
Kobarin 2.63a (FV) - In Japanese language only! Be sure to install Asian language packs for font support!
Nyam MP3 Player (late december 2014 build worked)
CoolPlayer 219 (June 1, 2008) (FV)
MPW (March 16, 2013) (FV) - Only 44.5kb in size; perfect for ancient machines ;)
yPlay (FV)
Ho MP3 Player 2.0 (FV)
VinAmp 1.5 (FV)
Diegote MP3 Player (FV) - Requires DirectX 9!
VU Player 2.49 (FV)
Apollo 37zz (FV)

Mounting Software
Virtual CloneDrive 5.5.2.0
Virtual Clone Drive 5.4.5.0???
**I still need to get around testing which versions do work.

Network

Office Suite
OpenOffice 2.4.3 (FV)
AbiWord 2.5.2 (FV)

Program Extractors/Hackers
IconsExtract 1.47 (CS?)

Registry Cleaner
Eusing Free Registry Cleaner 3.31 (FV)

Runtime Softwares
JDK 1.6.0 unofficial backport Win98 (FV)
Java 6u7 (FV) - Last version for Windows 95/98/ME
Flash Player 9.0.289.0  (FV)
Flash Player 10  (FV) - Unofficial backport

Screen Recorder
CamStudio 2.0 (FV)

Screensavers

SSH and Telenet Client
PuTTY 0.63 (test later versions)
 
Social Media Client
Skype 3.6.0.244 (FV)

System Profiler
CPU-Z 1.78 (FV)

System Utilities

Task Managers
TaskInfo 2000 3.0 (FV) - Still works on Windows NT 4.0. Use version 2.1 for a lighter footprint on system.
Sysinternals Process Explorer 11.11 (FV)
Revolutions Pack 9 (FV)

Text Editors
Notepad++ 5.9.1 ANSI (FV)
Metapad 3.6 (May 2011) (CS?)

Text-to-Speech Software
Microsoft Speech SDK 5.1 (Microsoft Sam/Mike/Mary) (FV)
AT&T Crystal (2008 or earlier build)
AT&T Paul (2008 or earlier build)
 
Video Editors

Video Game Engine

Virtualization/Emulation Software
Retro Arch - Windows 98SE/ME/2000 compatible build available.
Connectix Virtual PC 4.1
VirtualBox 1.5.6 (FV)
pSX 1.13 (FV)
VisualBoyAdvance 1.7.2 (FV)
Epsxe 1.9.25 - Works, but using save states on versions 1.6.0 (earlier untested) or later will crash the emulator. 1.9.25 cannot configure a CD drive. Using KernelEx makes no difference.
**Epsxe launches on a Vanilla Windows ME installation up to version 1.9.25. Version 2.0.0 will not launch with any KernelEx setting. I recommend using 1.9.0.
ZSNES

Windows Shells  
xplorer2 Lite (CS)
Meesoft Commander 1.36 (Requires GDIPLUS.dll)
File Navigator 1.7.4 (June 6, 2011)
Alt Commander 1.3-20071014 (FV?) - Compatible with Windows 95

http://wxglade.sourceforge.net/extra/cornice.html

Wireless Clients
Odyssey Client 4.52 (FV) - The only known working WPA2 wireless utility
Buffalo Wireless Manager V (FV) - Capable of up to WPA
Boingo Wireless (FV) - WEP only

Web Browsers
Opera 10.70 Build 9000 - DOSFreak's unofficial port with TLS 1.2 support
Retrozilla 2.1 - Windows 98/ME web browser with TLS 1.2 support; avoids being blocked out of modern sites.
Internet Explorer 6 SP1 (FV)
Netscape 9.0.0.6 (FV)

Windows Customization

Windows Patchers

Torrent Clients
uTorrent 1.8.5 (FV)
Vuze (March 2014 build)
 
Other
PunkBuster
abcAVI Tag Editor 1.81
DAV 2002 (mentioned by jumper???)
Randy Tab 1.1d - Guitar tablature authoring program

To Be Tested (Video Capturing)
Vidcap32
amcap
wmcap
vidcap (16-bit)