A holomorphic bijection from the open unit disc to the complex plane

By Liouville's theorem, there is no non-constant holomorphic function from the complex plane to the unit disc. I wonder what the converse is like--surely there are holomorphic functions on the open unit disc into the complex plane, but are there any bijective ones?

A bijective holomorphic function has a holomorphic inverse, so certainly there are no bijective holomorphic functions from $\mathbb D$ to $\mathbb C$.

For an example of a function form $\mathbb D$ to $\mathbb C$ that's unbounded, consider $\frac{1}{1-x}$ or $e^{\frac{1}{1-x}}$.

More generally: there is a holomorphic bijection from the open unit disk onto a region $U$ if and only if $U$ is simply connected and the complement of $U$ (in the Riemann sphere) has at least two points.