Simple example of idempotent but not commutative nor associative binary operator?

Question

Is there a simple example of a binary operation that is idempotent, but not commutative nor associative?

Answer 1

How about $a\oplus b=pa+(1-p)b$ for $a,b\in\mathbb R$ with $\{0,\frac12,1\}\not\ni p\in\mathbb R$?