My old answer: twice Despotism -> Monarchy -> Democracy
New new short answer: once Despotism -> Republic.
I almost never used Republic, and now love it. Income at 20% lux is nearly equal to what it would be with Monarchy and it's MP, when both are first obtained, and later on, I really have little to no problem going warmongering and staying in Republic. Late in the game Republic at 40% tax is still more income than Monarchy, and when peace comes my income skyrockets.
My long answer though.... it depends.
In some games, mostly deity, where I beeline Monarchy to have a chance to sell Polytheism (often neglected by AI), I still do two swaps. With a non-religious civ I'll almost never change three times, unless war weariness is overwhelming and I drop back to Monarchy.
In some games I can see that I'm going to need all the productivity out of my workers that I possibly can (maybe large jungles, or few workers and no native coal), which might push me toward making the jump to democracy. (The change in income between republic and democracy is negligible compared to the anarchy pain for a non-relig civ)
Charis