Too long for a comment. I fear I am far competent enough on the matter, yet I found the question very interesting and will attempt an answer for learning purposes, just to see where my understanding fails.
I was in fact rather sure that there cannot exist a function $f$ holomorpic on an open set $\Omega$ (which is a maximal connected open domain as in the OP's question) and continuous on $\partial \Omega$. In other words, only singularities can prevent further analytical continuation.
I would argue so.
As $f$ is continuous on the boundary of $\Omega$, one can find an open set $K$ on which $f$ is continuous, such that both $ K \cap \Omega \neq \emptyset $ and $K \cap \bar {\Omega} \neq \emptyset$ hold.
Let then define as $M$ any open set contained in $K \cap \Omega$.
The the Laplace equation on $M$ will be solved by a function $g$, coinciding with $f$ on $M$. As $f$ is continuous on $K$, I believe $g$ can be extended at least to $\partial \Omega$ as well.
Would not then $g$ constitute an analytical continuation of $f$ at least up to $\partial \Omega$, contrary to the assumed "maximal domain" hypothesis?