operation that is commutative but not associative 
Does there exist an operation that is commutative but not associative?

do you know what is operation? 

Take for instance $f(x,y)=2x+2y$ on $\mathbb{N}$. Then $f(x,f(y,z))=2x+4y+4z$ and $f(f(x,y),z)=4x+4y+2z$.
Another example is $f(x,y)=2^{x +y}$ on $\mathbb{N}$. Then $f(x,f(y,z))=2^{x + 2^{y + z}}$ and $f(f(x,y),z) = 2^{2^{x + y} +z}$.
