I've been on a few other boards that have posting games (some even have a separate section for them) but I don't see any here.
A few examples are: Answer a question with a song title, The corrupted wish game, This or that, Various alphabet games, Post a useless rumor about the person above you, Word association games, etc... Basically they're good for pointless banter and good fun. Sometimes when site activity is slow, they keep people entertained until posting picks up.
Would they be against the rules in the Off-Topic forum? If so, could you add a sub-forum for them??