I am working on a question that requires me to prove that *on every simply connected open subset of $\mathbb{C}$, there exists a holomorphic function that cannot be extended to a holomorphic function on a larger connected open set.*

I know an example of such a holomorphic function on the open unit disc: $$f:z\mapsto\sum\_{n=1}^\infty z^{n!}.$$ I tried to combine this with the Riemann mapping theorem.

Let $\Omega$ be a simply connected open subset of $\mathbb{C}$ and one may assume that $\Omega$ is not the entire complex plane. By the Riemann mapping theorem, there exists a conformal equivalence $$\phi:\Omega\rightarrow D$$where $D$ is the open unit disc. Then my guess is that $f\circ\phi$ has no analytic continuation, but then I had to link the boundaries of $\Omega$ and of $D$ and I don't know what is going on between $\partial\Omega$ and $\partial D$.

Could anyone offer any idea? Many thanks!