The most promotions you can get (And the AI can get) is (with Unrestricted leaders):
Agg/Pro of Native America
Your archers start with:
Combat 1
Drill 1
CG 1
Then... after that, you have the totem pole (+3 XP for Archers), Barracks (+3 XP), and Vassalage and Theocracy (+2 XP, and another +2 XP with state religion)
That comes to 10 XP exactly... enough for *3* promotions.
So Tokogawa of Native America's longbows can theoretically start with Combat I, Drill II, CG IV ... Combat I, Drill IV, CG I ... Combat V, Drill I, CG I ... so many options! And that comes to 6 promotions in total.
So yes, you can get quite a bit of XP without warring.
Without Unrestricted leaders, the maximum is 5. Barracks + Stable = 2 free promotions (+5 XP), as does Barracks + either Vassalage or Theocracy (with state religion), and then protective gives 2 free promotions, and aggressive gives another 1. So Tokogawa can get 5, and Sitting Bull of the Native Americans can also get 5 (Barracks + Totem Pole + Vassalage + Theocracy gives +10 XP = 3 promotions, and Sitting Bull is also protective, giving Drill I and City Garrison I for free)