Composition of Formal Power Series is associative
Solution 1:
A formal power series $g(x)$ such that $g(0)=0$ gives you a continuous ring homomorphism $g^*:K[[x]]\to K[[x]]$ uniquely defined by $x\mapsto g(x)$. In fact, $g^*(f)=f\circ g$ (it is clear if $f$ is a polynomial and extends by continuity to power series). As $h^*(g^*(x))=h^*(g(x))=g(h(x))=(g\circ h)^*(x)$, we have $h^*\circ g^*=(g\circ h)^*$. Evaluating both sides on $f$ we get $(f\circ g)\circ h=f\circ (g\circ h)$.