I've consistently used the same character for the majority of my Rps. Her personality and appearance are the only thing that stays the same, though. 
She's existed as a Cyberpunk cop, a Victorian nanny-turned mage, D&D mage, mercenary, fantasy mercenary, star trek-esque medic, etc.

In each iteration, I've adapted her slightly and added a bit more or altered her mythos slightly. There was a long streak of her being a musician, yet since it was so inconsequential on the RPs I played, I dropped it. She's often been a medic or another 'healing' role, which I have also been moving away from. If there are guns, she'll most likely have an uzi, and if there's magic, it's likely that she's a pyromancer. But of course, I change what is needed for the game.

So it's less like me shoe-horning a single character into every single game, it's more like infinite alternate realities of the same woman.
Maybe I'll have a crossover RP that I host, where everyone plays as an iteration of her, and they gang up and take down the big bad, who is a lazy RPer who likes to play as what he knows.