Q: Is there a binary operation $*$ such that $(a*a)*a\neq a*(a*a)$?
I apologize if this question has been asked before or maybe it is too stupid. It is not obvious to me that any binary operation always must satisfy $(a*a)*a= a*(a*a)$. I tried to find counterexamples, but so far no luck.
Certainly this binary operation, if it exists, is not associative.