A particular holomorphic function on a hyperconvex domain.

Question

Let $X$ be a Stein manifold and $\Omega\subset X$ be hyperconvex, i.e. there exists a negative plurisubharmonic exhaustion function for $\Omega$. Since $\Omega$ is hyperconvex it is a domain of holomorphy so in particular the following holds: For any $a\in \partial \Omega$ there exists a function $f$ holomorphic on $\Omega$ which has no holomorphic extention to any neighbourhood of $a$. My question is as follows: Can I construct $f$ such that $\lim_{z\to a, z\in \Omega}|f(z)|=\infty$?