Morrowind:Arch-Mage

Quick Walkthrough
Peacefully:
# Talk to Skink-in-Tree's-Shade in the Wolverine Hall Mage's Guild for duties.
# Travel to Vivec to deliver the Letter from Ocato to Trebonius Artorius.
# Give the letter to Trebonius in the Foreign Quarter Guild of Mages and be promoted to Arch-Mage.
Duel to the death:
# Either talk to Edwinna Elbert in the Ald'ruhn Guild of Mages to be directed to Trebonius or talk directly to Trebonius about becoming the Arch-Mage.
# Trebonius will challenge you to a duel to the death in the Arena.
# Defeat Trebonius to become Arch-Mage.

Detail Walkthrough
Becoming Arch-Mage
In order to achieve your last guild rank of Arch-Mage you must replace Trebonius Artorius in the Vivec Guild of Mages as the guildmaster. This isn't much of a surprise if you're familiar with Trebonius; he seems a bit light headed to be head of the Guild. There are two ways of doing this: peacefully or by a duel to the death.

Rising to Arch-Mage Peacefully
Once you've finished the Galur Rithari's Papers quest for Skink-in-Tree's-Shade (in the Wolverine Hall Mage's Guild), and reached the rank of Master Wizard, talk with Skink to be given a letter from Ocato (spelled correctly this time) that amounts to Trebonius's pink slip (he's being fired by the Empire). In it, Ocato asks Trebonius to promote you to Arch-Mage. Trebonius will, but he won't step down; you end up both being Arch-Mages, and you get +2 Reputation bonus. The downside is the only way you can get his equipment is by doing the Dwemer and Telvanni quests he sets for you.  Because of how the journal script progresses, once you have the letter, you won't be able to challenge him to a duel unless you have previously done so.

If you do deliver the letter from Ocato and peacefully become Arch-Mage, you can still turn in Trebonius's quest Mystery of the Dwarves and receive his point of reputation reward. After which there is no additional value in keeping Trebonius alive. Using Taunt or Frenzy he can be fought and killed inside the Vivec Mage's Guild. This gives access to his equipment AND the maximum Reputation gain. Other NPC's and quests reference his death as if you had fought in the arena except you gain the 3 reputation points from the Mystery of the Dwarves and delivering Ocato's letter without having to sacrifice the valuable Necromancer's Amulet or killing the Telvanni Counselors.

Duel With Trebonius
Similarly, if you have finished the Bethamez quest for Edwinna Elbert (in the Ald'ruhn Guild of Mages), and reached the rank of Master Wizard, talk to Edwinna about duties and be directed to Trebonius in the Vivec Guild of Mages. He will greet you with a challenge to a duel to the death in the Vivec Arena.

Alternatively, if you have reached the rank of Wizard in the Guild, you can talk directly to Trebonius about becoming the Arch-Mage. He will again challenge you to a duel to the death in the Arena.

Either way, if you can manage to defeat him, you'll become the new Arch-Mage. You'll also get to keep his great equipment, but don't get any reputation points.

Notes
If you kill Trebonius in the Arena but forget to remove his unique items, those items will appear in the chest in the downstairs office of the Arena, Fighter Training, if you have the patch.
Killing Trebonius will prevent you from completing A Rash of Insults, so it is best to complete that quest before killing him.

Bugs
 
 
 If you have access to the console you can fix that by opening it and entering the command:  
 
  
 Use the following console commands to rejoin:
::PCClearExpelled - Confirmed as not working
::Set ExpMagesGuild to 0

Quest Stages