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@alcinnz A smallest counterexample is, apparently, the "winner of rock paper scissors" operator (over the three-element "rock paper scissors" set), which is commutative but not associative.

I'm not quite convinced (yet) that it's unique up to isomorphism, but Wikipedia implies it is, and who am I to doubt uncited passages in Wikipedia articles about abstract algebra?

Edit: never mind. It's cited. https://doi.org/10.1016/j.ic.2014.08.005