jmc

Consider two arbitrary polynomials $p(x)$ and $q(x)$ with highest degree terms $x^n$ and $x^m$, respectively. Then $p(q(x))=(q(x){)}^n+\cdots =(x^m+\cdots {)}^n+\cdots =x^{mn}+\cdots$ is a polynomial of degree $mn$. It follows that $f(g(x))$ is a polynomial of degree $18$. Then, either $g(h(x))$ or $h(f(x))$ must be a polynomial of degree $36$. This gives that the degree of $h(x)$ is either $12$ or $6$, but in the former case, the degree of $h(f(x))$ would be $72$. Thus, the degree of $h$ is $\boxed{6}$.