What kinds of problems are still fundamentally hard for LLMs?

I played a game of a custom chess variant against an claude opus 5.5, and it beat me.
The game combined several rule changes: the board wraps around from the h-file to the a-file, knights move three squares in one direction and one sideways, and captured pieces can be dropped back onto the board, as in crazyhouse. I gave the model the rules, the starting position, and a board diagram, then asked it to reply with one legal move at a time.
Also I recreated this board game https://nika-game.com/ and played with claude, and it still win, although I believe it is really unpopuplar and old game without much training data available (claude didn't even know the rules initially)

I used to think chess exposed a fundamental limitation of LLMs: keeping track of a changing board, following exact rules, and planning ahead seemed like a poor fit for a language model. This game made me reconsider that assumption.
So I’m curious: what broad classes of problems do you think LLMs still can’t solve reliably? Are there limitations you consider fundamental/archiectural, rather than problems that might improve with better models, more computation, or tools? What would be a good test?