The only way to gain levels while playing Idle (assuming no Arenas) is via Achievements. I don't think birthday achievements go past 2 or 3 years in; I certainly got no points for my Fifth Birthday.
So you just need to add up the points for all the Achievements you can get without playing Classic and see what level that gets you to.
EDIT: I see JK already did that. Going by levels then, Idle alone can get you to level 21 and almost 22 (https://www.warzone.com/wiki/Levels
Edited 4/5/2021 21:54:24