TITLE: Surjective entire functions without critical points
QUESTION [7 upvotes]: It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk. 
I am pretty sure that the answer to the following is positive and well-known (to experts) and is somewhere in the literature:
Question. Are there (non-affine) surjective entire functions $f: {\mathbb C}\to {\mathbb C}$ without critical points? 
Such functions (up to an additive constant) would have the form $f(z)=\int_0^z e^{h(w)} dw$, where $h(w)$ is another nonconstant entire function. I do not see, however, how to verify surjectivity of such $f$. Of course, it can miss at most one point in ${\mathbb C}$ and, hence, it would be astounding if indeed this was the case for arbitrary $h$. However, I do not see how to rule this out. Applying Cauchy's argument principle in this setting looks extremely messy.

REPLY [7 votes]: The simplest surjective entire function without critical points is the "probability integral", $\int_0^z e^{-\zeta^2}d\zeta$.
Proof. Suppose it omits $a$. As it is of order 2, it must be of the form $a+e^P$,
where $P$ is a polynomial of degree $2$. But this function has critical points:-)
The argument extends to $\int e^Q$ for any polynomial of degree at least $2$.