Suppose we have a function $f(x) = \frac{h(x)}{p(x)}$ where $h(x)$ and $p(x)$ are polynomials. Can any iterative composition of $f(x)$ like $f(f(x))$ or $f(f(\dots f(x)\dots)$ be a polynomial?
Update
Let me add a few more constraints. $\deg(p(x)),\deg(h(x)) \geq 1$ and $p(x) \nmid h(x)$
Furthermore if $\dfrac{h(x)}{p(x)}$ can be reduced, then neither the numerator nor the denominator is ever a constant, i.e the degrees of the numerator and denominator are always greater than or equal to $1$
For example, if $f(x)=1/x^n$, then $f^2(x)=x^{n^2}$