I've always wondered ever since I read Dark River games that cats played. We've already seen "Warriors" in Bluestar's Prophecy and play-fighting/hunting in almost every other book. And DarkClan, of course!
I've always thought the kits had games similar to Hide and Seek (I'd imagine they'd do it a lot in Greenleaf or Leaf-fall since around that time of year there's a lot of stuff to hide in) or Chase/Tips/Tag. (Some people call it Chase, around where I live we call it Tips or Tag. Basically that game where one person runs around and tries to tag their friends.)
Also, I think Clans would happen often. (In fact, many of my Clan OC's had played Clans. The OC I'm currently building for RP here loved to play as Stonestar, leader of uncreatively-named StoneClan, with his sister being his loyal deputy, Featherbrook as a kit. They also bothered their irresponsive brother, pretending he was medicine cat when all he wanted to do was not be involved in anything.)
Maybe they had something like "Capture the Flag" with prey before that thing with Driftkit and the bird of prey I cannot remember in Code of the Clans? They might still play it now, but with a leaf or stick or something.
What are your thoughts? Do the kits in your fanfiction play other types of games? What are they?