I play on Farm Town, FarmVille, Pet Dragons 1, and Pokemon Adventure.
Farm Town and FarmVille are quite addicting. Because you've got to be into the game to get anything out of it...as the more you plow, plant, harvest and sell, the more experience and money you get.
Pet Dragons 1 is just there as a time waster. I often forget I have it so my dragon is old, but not very strong.
Pokemon Adventure was a huge hit on Facebook last summer, when it was released. It is kind of a sequel to the "Send Pokemon" and "Send Pokemon: Johto" and certainly an upgrade, becoming an MMORPG. I went through a chat interview process and became a moderator at their chatrooms and later at the IRC. Come October, though, the creator of PA became too busy with life and stopped running the app, leaving its fans hanging in Pewter. Today it stands dead, never more than 15 people online at one time, when there used to be 1,000-2,000+ people on at any given time.
So yeah, those are the games I play on facebook. Share your stories of Facebook game apps