One of my favorite things to do in Civilization is to play on a small map on one of the lower difficulty levels and see if I can build all of the wonders in the game. (Not take through conquest, but actually build myself.) I first got into this when playing Alpha Centauri. I haven't accomplished it with Civilization V just yet. I came close, but so far I've always managed to miss one.
Also, the "One City Challenge" is pretty fun in this iteration of the game. And not as difficult as it sounds. In general the game has been balanced so that empire size is no longer the most important attribute of your civilization. You can run a small empire and still do really well.
This game is so fun it offends me.