Since $S$ is finite, there is a finite set of all functions $\left\{g_{1}, g_{2}, \ldots, g_{k}\right\}$ from $S$ to itself. Consider a function $F$ that assigns to each positive integer $n$ one of these functions such that $f$ is the $n$-th power of the function $F(n)$. So $F$ induces a partition of the set of all positive integers into sets $P_{i}$ consisting of all the integers $n$ such that $F(n)=g_{i}$.
For any positive integer $N$, consider the complete graph $K_{N}$ on $N$ vertices labeled 1 through $N$. We will colour the edges of $K_{N}$ in $k$ colours $C_{1}, C_{2}, \ldots, C_{k}$ according to the partition in the following way: If $|x-y|$ lies in $P_{i}$, colour the edge between $x$ and $y$ in the colour $C_{i}$. By Ramsey's theorem we can take $N$ to be large enough that there is a monochromatic triangle in $K_{N}$. This means that there are three integers $x, y$ and $z$ and an index $i$ for which $|x-y|,|y-z|,|z-x| \in P_{i}$. Hence there are three integers $a, b, c \in P_{i}$ such that $a+b=c$.
Therefore, some function $g_{i}: S \rightarrow S$ satisfies $f(x)=g_{i}^{a}(x)=g_{i}^{b}(x)=g_{i}^{a+b}(x)$ for each $x \in S$. Hence, $f(f(x))=g_{i}^{a}\left(g_{i}^{b}(x)\right)=g_{i}^{a+b}(x)=f(x)$ for each $x \in S$.
Remark: The fact that there is an index $i$ for which $P_{i}$ contains three integers $a, b, c$ such that $a+b=c$ is known as Schur's theorem.