I didnt found any hint in 5th Edition.
From 3rd Edition I just remember: a Magus have to challenge an Archmage, the Archmage sets a contest. If the magus win, he get the title, the old archmage loose it ... which would lead to lesser and lesser archmages in the order and to they question: how they first archmages became archmages?
Archmages in 5e are described in Guardians of the Forest p24.
To summarise, to become an Archmage you need to invent a spell of 7th+ magnitude, be famous for some great deed, and win a challenge against an existing Archmage. Each existing Archmage can grant the title to at most seven challengers.
