In a way every game is a rationality game, because in almost every game you have to discover things, predict things, etc. In another way almost no game is one, because domain-specific strategies and skills win out over general ones.

One idea is based on the claim that general rationality skills matter more when it's a fresh new game that nobody has played yet, since then you have to use your general thinking skills to reason about things in the game and to invent game-speficic strategies. So what if there were "mystery game" competitions where the organizers invented a new set of games for every event and only revealed them some set time before the games started? I don't know of any that exist, but it would be interesting to see what kinds of skills would lead to consistent winning in these competitions.

There are various other ways you could think of to make it so that the game varies constantly and there's no way to accumulate game-specific skills, only general ones like quick thinking, teamwork etc. Playing in a different physical place every match like in HPMoR's battles is one.