There are a few games (DotA, TF2, L4D) that I play interchangeably (albeit it some more than others) with friends whenever we feel like it (and can be considered infinitely replayable). On top of that I usually have at least 1-2 games on console that I go to every so often. Right now it's Okami for the Wii and Fallout 3 for the 360. Of course, the latter is limited since I don't have a 360 and my friend that does shares it with his brother, so he only gets it 2 weeks at a time <_<. I also have HL2 that I never beat that I'll usually go back to and play for an hour or so, but I do that once every other week or so
. And then there's my DS, which always has a DS game and a GBA game in it, although I can't say I've really touched the thing in about a month (stupid school)...
So yes, I can have anywhere between 3-8 games unfinished at a time, depending on if you count games that you never really "finish".