One game I play is Stepmania, which is a DDR (Dance Dance Revolution -- a
dancing game) simulation. This probably isn't a good one to get in to because
you need to download songs and stuff for it. But if you can find them, the
game is free. The license gives you permission to use, modify, and distribute
it as long as you include the license. (It is not GPL but appears to follow
the 4 freedoms.)
You can find a whole list of potentially-free games if you go here and in the
search terms select Distribution Policy = "Freely Distributable" and select
Source Code Available = "Yes." Then if you find something you like, download
the source code and read the license to make sure it provides you with the 4
freedoms (0: freedom to use, 1: freedom to modify, 2: freedom to distribute
original code, 3: freedom to distribute modified code) and that if you do
distribute it you are required to keep its license.
Keep in mind even if you find a game like that (I'm sure there are plenty)
you may still not be able to use it freely if it requires non-free libraries.
Hope that helps!