Operator which is symmetric but not associative? Addition and multiplication are symmetric and associative. But I have no idea about operators which are symmetric but _not_ associative. Please help me listing any such 2-arity operators.

Symmetric operations are easy to come by, aren't they? Just write down a symmetric polynomial at random, like $x^2+y^2$, or make up a symmetric operation table haphazardly. It might be associative, but the odds are it's not.