Over the course of time, I usually end up with both "versions" of the generations that come out (X and Y, OR and AS, etc). Right now, Y is my big huge storehouse, breeding center, and EV training hub, and remains untouched. However, a recent hobby of mine has been to restart on another game - usually of the latest in the generation - with bred 5 IV and EV trained members of my currently successful or most fun Showdown OU teams. Right now, I just restarted with Treecko, and have bred good Gligar, Marill, and Scyther to add on later. I try and set rules for myself on the playthrough, like I'm not allowed to add the Marill until I could feasibly catch on in-game, and I'm not allowed to add Scyther until I have access to the first 500+ BST Pokemon. I'm weird, but it's great fun, and really gets me more attached to the members of my current team (especially with Amie now). It's also a more fun way to train them to level 50 so I can use them in battles and online, and have fully competitive thing I am doing the playthrough with.
I always keep one game as a storehouse, though. I still have LOTS of thing on Diamond I should probably transfer over, for instance. That's just sooo much time...