We show that the division operation over nonzero reals is non-associative. All we need is a counter-example: so let us compare $1/(1/2)$ and $(1/1)/2$. The first expression is equal to $2$, the second to $1/2$, hence division over the nonzero reals is not associative.

Remark. The property of being associative of a binary operation can be generalized to an arbitrary $n$-ary operation, where $n\geq 2$. An $n$-ary operation $\phi$ on a set $A$ is said to be associative if for any elements $a_{1},\ldots,a_{{2n-1}}\in A$, we have

In other words, for any $i=1,\ldots,n$, if we set $b_{i}:=\phi(a_{1},\ldots,\phi(a_{i},\ldots,a_{{i+n-1}}),\ldots,a_{{2n-1}})$, then $\phi$ is associative iff $b_{i}=b_{1}$ for all $i=1,\ldots,n$. Therefore, for instance, a ternary operation $f$ on $A$ is associative if $f(f(a,b,c),d,e)=f(a,f(b,c,d),e)=f(a,b,f(c,d,e))$.