Are all associative operations essentially function composition? It is well known that function composition is associative. Is the converse true? Is any associative operation essentially expressible as function composition? If not, what's an interesting example of an associative operation that is not expressible as function composition?

Note that any operation $\star :S\times S\to S$ is equivalent to a function $f: S\times S\to S$ defined by $a\star b=f(a, b)$.  We could even denote $f:=\star$, so the difference between $\star(a, b) $ and $a\star b$ is purely notation based (known as prefix vs infix notation respectively). 
For a given associative operation $\star$,  we have that $$a\star (b\star c) =(a\star b) \star c\iff f(a, f(b, c)) =f(f(a, b), c)$$
In this way, each binary operation $\star: S\times S\to S$ being associative can be thought of being similar to function composition being associative. Also, the operation being from $S\times S\to S$ is important to the above. If there's some form of associativity that doesn't depend on this, you may get another answer. 
