I don't know how long these have been around (probably not very long since they're still in beta) but apparently Squenix has released two Facebook games. Knights of the Crystals (an RPG where all of the dungeon crawling and most of the battles take place off screen, for serious) and Chocobo's Crystal Tower (which is about raising Chocobos.)

I'm not particularly impressed by either of them, I have to say. :P