You could train an AI just to play chess. Sites like chess.com have tens, hundreds of millions of games to use as training data. But the AI isn't "thinking" though, it's just being asked given this input, what's the most likely outputs and picking one based on its settings. Then the other player moves, the context updates, rinse, repeat. Such an AI would likely whoop most people's asses but experienced players might figure how to lead it down a path where it doesn't sufficient training data to play strongly.
But it's not a generalized LLM like ChatGPT where it's picking up a handful of chess games from god knows without knowing or enforcing the rules or anything else.
Likewise I bet we'll see AIs for poker and other lucrative online sports. I bet a lot of online casinos have amassed huge stores of data to produce AIs, as well as players using scraping or logs to do the same. I could even see online casinos running AIs in games because it's a way of taking money from players beyond the normal rake.