Is there a characterization of entire functions with image $\Bbb C \setminus \{0\}$?

Question

Do we have a characterisation of entire functions with image $\Bbb C \setminus \{0\}$?

If not, is there an example of such a function that is not in the form of $\exp(g)$ for some entire function $g$?

Answer 1

Hint: If $f$ is never zero in $\mathbb C$, then $\frac{f'}{f}$ is entire and so has a primitive.

Answer 2

If $f$ is entire and $f(z) \ne 0$ for all $z$, then there is an entire $g$ such that $f=e^g$, since $ \mathbb C$ is simply connected.