There are lots of operations that are not commutative.
I'm looking for striking counter-examples of operations that are not associative.
Or may associativity be genuinely built-in the concept of an operation? May non-associative operations be of genuinely lesser importance?
Which role do algebraic structures with non-associative operations play?
There's a big gap between commutative and non-commuative algebraic structures (e.g. Abelian vs. non-Abelian groups or categories). Both kinds of algebraic structures are of equal importance. Does the same hold for assosiative vs. non-associative algebraic structures?