Experienced with the game, but not necessarily with the Inca - I've played them once, Mesix appears only to play Babylon except when experimenting with other civs to establish that they aren't as strong as Babylon...
Actually I rarely play Babylon. I played them a few times and discovered how much faster they were to tech (on account of having essentially double the

of any other civ in the early game when it is the most crucial). I played my first G&K Babylon game just last week when I started to compare if Korea could indeed keep up with them (they can't, and no one has really demonstrated that they can except with empty hyperbole). I have also been experimenting with some different strategies with them (like a OCC game and delaying expansion until the Renaissance Era, both of which were fun).
Since G&K came out, I have mostly been playing the scenarios and the new civs which came with the expansion (trying to unlock all the achievements). I really like the Into the Renaissance scenario. Before G&K, China and Egypt (until I was able to kick my wonder addiction) were the civs I played the most often. I used to rush the AI Babylon in games to capture their capital with the Academy early in the game, but they don't seem to favor planting the Academy any more (perhaps patched in the expansion?).
The one thing that has come out of all this discussion is that I played around with Maya and Korea more. I am trying to find the magic that makes everyone love Korea so much, but it just isn't there for me (which is strange because I love to play a specialist economy game). Despite the fact that I win most games by domination and quite enjoy being a warmonger, I consider myself to be more of a builder and prefer to play civs which have UB instead of two UU or (even worse) a UI.
While testing the Maya, I have found that the downside of increase the cost of future great people really does stop them from being overpowered. Sure, you can't game the system to get 2 great people on 1 turn, and the GS won't come as early, but Maya generates quite a lot of great people and can also get religion easy enough. Here is a list of the great people which I got in the first 175 turns playing as Maya: GS (72, UA), GE (86, UA), GS (89, Liberty), GP (89,

) GP (101, UA), GA (117, UA), Adm (133, UA), GM (152, UA). That is eight great people in the preindustrial era, and I know that with practice I can probably get a couple more. The one thing I would mention is that the great prophet from the Long Count increases the cost of future great people generated in cities and also future Great Prophets generated with

. When playing Maya, I would recommend generating a Great Prophet with

to start the religion and then taking the "free" Great Prophet from the next Long Count to enhance the religion.