The size of $|p^n(a)|$ can be very different depending on $a$. If $a$ is periodic (in particular, fixed) point of $p$, this sequence will be bounded. If $a$ is in the domain of attraction of infinity then $\log|p^n(a)|$ is like $d^n$ where $d$ is the degree.
More precisely, for each polynomial there is a compact set $K(p)$ in the complex plane, called the filled in Julia set, such that for $a\in K(f)$ your sequence is bounded, while for $a\not\in K(p)$ it grows like $d^n$. When searching on Google look on "Holomorphic dynamics".
Remark. It is irrelevant for what I wrote above, whether $a$ is an integer or not and whether the coefficients are integer or complex. However, as Joe Silverman suggested in his comment, if the coefficients and $a$ are integers, then $p^n(a)$ are also integers, and as $K(f)$ may contain only finitely many integers, the sequence can be bounded only if it is pre-periodic, that is becomes periodic after the removal of finitely many terms.