Me and a friend of mine didn't manage to solve the following problem.
Let $f: \mathbb{C} \to \mathbb{C}$ be an intere holomorphic function having a finite number of zeroes. Then either $f$ is a polynomial or there is a succession $\{z_j\}$ such that $z_j \to \infty$ and there exist $r$ such that eventually $$|f(z_j)| > e^{r|z_j|}. $$
Attempts
Let's call $h = \frac{f}{g}$, where $g$ is the polynomial that vanishes on zeroes of $f$ with the same multiplicity of $f$.
We tried to look at $\frac{h'}{h}$, the logarithmic derivative of $h$ , but without good ideas.
One can observe that $h$, when it's not constant, must have an essential singularity at infty.
If $f$ is not a polynomial, then $h$ is a nonconstant entire function without zeros. Therefore, $\phi = \log h$ is a nonconstant entire function. So its derivative at some point $z_0$ is nonzero. By the Cauchy integral formula, $$ \phi'(z_0) = \frac{1}{2\pi i}\int_{|z-z_0|=r} \frac{\phi(z)}{z^2}\,dz $$ which implies that $\max_{|z-z_0|=r}|\phi(z)| \ge r|\phi'(0)|$. Translated back in terms of $h$, this yields $$ \max_{|z-z_0|=r}|h(z)| \ge e^{r|\phi'(0)|} $$