Suppose $f$ is an analytic function with power series expansion $f(z)=\sum_{n=0}^{\infty} a_nz^n$, and $p = \sum_{n=0}^{d}b_nz^n$ is a polynomial. If $f$ is a polynomial of degree larger than $d$, then $|f|$ grows faster than $|p|$, but the situation is not so clear when the expansion of $f$ has infinitely many nonzero coefficients. I would expect the growth of the function $f$ then to be faster than that of $p$, as with the function $e^z = \sum_{n=0}^{\infty}\frac{z^n}{n!}$. However the function $\frac{1}{1-z} = \sum_{n=0}^{\infty}z^n$ also has infinitely many nonzero coefficients and grows slower than any polynomial (as $|z|\to\infty$). I realize this is related to the failure of the power series to converge outside a disk of radius $1$. Also, $log(z)$ grows slower than any polynomial, but any power series representation cannot converge on an infinite radius (The function itself cannot be well-defined everywhere in the complex plane simultaneously).
Under what conditions can we say that a power series with infinitely many nonzero coefficients represents a function that grows faster than any polynomial? Is this true for any power series with infinite radius of convergence? Are there such power series which grow at the rate $z^\alpha$, for any $\alpha\in(0,\infty)$?
I have in mind the case where $f$ is complex-analytic, but I would also be interested to hear about the case where $f$ is real-analytic, if the cases differ.
