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?

Assume we have a set $X$ and an associative binary operation $*$ on it. Let $\bar X=X\cup\{\perp\}$ where $\perp$ is a symbol $\notin X$.
Let $Y=\bar X^{\bar X}$ be the set of maps from $\bar X$ to itself. 
Of course $Y$ is a monoid under function composition.
We have a map $\Phi\colon X\to Y$, given by 
$$\Phi(x)\colon z\mapsto\begin{cases}x&\text{if }z=\perp\\x*z&\text{otherwise}\end{cases} $$
Note that $\Phi$ is injective because of the $\perp$ case.
Note that $$(\Phi(a)\circ\Phi(b))(z)=\begin{cases}\Phi(a)(b)=a*b&\text{if }z=\perp\\a*(b*z)&\text{otherwise}\end{cases} $$
Because of the associativity of $*$, we see that
$$\Phi(a)\circ\Phi(b)=\Phi(a*b),$$
i.e., $\Phi$ is an injective monoid homomorphism from $X$ to $Y$.
In other words, we can identify the elements of $X$ with certain functions in such a way that $*$ becomes function composition.
