I wondered about the following some time ago:
Let $\Omega \subsetneq \mathbb C$ be a domain. Let $\psi_n: \Omega \to \Omega$ be a sequence of biholomorphisms converging to some $\psi$ locally uniformly on $\Omega$.
Is $\psi$ necessarily surjective?
Some observations:
We have $\psi_n(z) = \frac{z}{n}$ as a counterexample on $\Omega = \mathbb C$.
There is a biholomorphism $\phi$, which maps $\Omega$ into the unit disc $\mathbb D$. So considering $(\phi\circ \psi_n\circ \phi^{-1}): \phi(\Omega) \to \phi(\Omega)$ we can reduce the general case to the case of $\Omega \subset \mathbb D$ being bounded.Assuming $\Omega$ to be bounded:
The derivatives \psi'_n of $\psi_n$ also converge locally uniformly to \psi', so $ \begin{align} \mu(\psi(\Omega)) &= \iint_{\Omega} |J_{\psi}(z)| \; \mathrm dx\,\mathrm dy \\ &\ne \lim_{n\to \infty} \iint_{\Omega} |J_{\psi_n}(z)| \; \mathrm dx\,\mathrm dy \\ &= \lim_{n\to \infty} \iint_{\psi_n(\Omega)} \; \mathrm dx\,\mathrm dy \\ &= \mu(\Omega) \end{align} $ i.e. $\psi$ is 'almost surjective'.
I don't know how one might proceed from here (I hope I haven't made a mistake in my obeservations).
I'd be interested to see an answer to this question. =)