I might be stretching the question to a poll I was personally hoping for, as I am reading more into the "favorite" part than the "best" part, as best has been debated over and over again. And by "favorite", I'm stretcing that into "most fun".
Personally, I'm making a point for De Gaulle from France. France's UU and UB are not good (well, the musketeer is kind of fun if you get to gunpowder early and the path to rifling is still kind of long...), though the nice starting techs are very relevant, but the real joy here are De Gaulle's traits.
Come on, I know it's not the most powerful combination at all, but Industrious to wonder-whore and Charismatic to play with all those promotions has got to make for the most fun "recreational" Civ experience!
Most of my games are started with random leaders (and most of the time, unrestricted leaders), but from time to time, I start up a De Gaulle game, just because it's so much fun! (and don't get me started on the unrestricted leaders options, with De Gaulle for the fun traits and the civ of your choice for fun UUs and UBs...

)
PS: If I need to go back to "best" civ, discounting leader traits, the Incas are a staple of course and pretty obvious, but let's get some love for the Chinese. Starting techs are good (and fun because never frustrating), and the Cho-Ko-Nus are a very interesting UU. They are really powerful, but they need some commitment and some plan to make good use of them. You can't just prey, like with War Chariots, that there'll be some nearby horses and blindly rush into the nearest civ. The Cho-Ko-Nus require some planning and are stil early enough that they can be relevant in any game.
I have to say, any Spiritual civ can be really fun too! It opens so many options, it makes for interesting games and mechanics.
Edit: So much love for De Gaulle, but still messed up my original post by mixing him up with Napoleon...

Napoleon is fine but not really fun... Go De Gaulle
