Is there a $2\pi i$-periodic holomorphic function $f$ on the complex plane $\mathbb{C}$, $$f(z+2\pi i n)=f(z), \, \forall n\in \mathbb{Z} \, \forall z\in\mathbb{C}$$ that blows up in both directions of the real axis, $$|f(z)|^2 \to\infty$$ as $z\to+\infty$ and $z\to-\infty$ and whose derivative has no zeroes, i.e. $$f'(z)\neq 0$$ for all $z\in \mathbb{C}$?
I cannot find one, and my guess is that such a function does not exist. If I am correct, how can I proof it? If I am incorrect, what is an example for such a function (even better would be an iteration/classification of all such functions)?
Edit: I tried to make the divergence-requirement more precise.
Assuming that you want $$\lim_{\operatorname{Re}(z) \to \pm \infty} |f(z)| = \infty,$$ there is no solution; translating the problem via the exponential function into the corresponding problem of finding a function on $\mathbb C^\times$ which approaches infinity at both $0$ and $\infty$, that means the singularities at $0$ and $\infty$ are not essential, and therefore they must be poles. Which makes $f$ a meromorphic function, so $f'$ is also a meromorphic function, with poles of one degree higher at $0$ and $\infty$. Since meromorphic functions on the Riemann sphere have the same number of zeroes as poles, $f'$ therefore has a zero.