What (if any) games are unsolvable?

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Some games have been solved (checkers, tic-tac-toe), some games can be solved (Chess, Go); what I want to know is: are there any game which cannot be solved?

Supposing you had an infinite amount of computing power and could writing an infinitely large book containing all games states, are there any games which would still be unsolvable? Is it just any game with chance (cards, dice, tiles etc)? I’ve heard backgammon has been solved though, so maybe it’s more complicated than that.