Which Stack Exchange site is appropriate for such a question?
None currently. I had a proposal in Area51 for it but it appears to have been closed.
I recommend Gaming's Chat, there are (mostly) nice people that love to talk about games there.
If you mean video games, there is a StackExchange site where this is on-topic. It's called Software Recommendations. Video games are software—so, yeah. There is a games tag. See the help center for more information on what kind of game recommendation questions you can ask.
Anyway, if you also want to ask about board games, pen-and-paper RPGs, social games, night games (e.g. capture the flag) and stuff, I've begun looking for such a site—but, I haven't found one, yet.