Suppose a function $f$ is holomorphic in a punctured neighborhood of a point $z_0 \in \mathbb{C}$. Suppose that for all fixed $\theta \in \mathbb{R}$, $\lim_{r \to 0} |f(re^{i\theta})|=\infty$. Does it necessarily follow that $f$ has a pole at $z_0$? Or is it possible for the singularity to be essential?
(Recall that if $\lim_{z \to z_0} |f(z)|=\infty$, then $f$ must have a pole. But I'm requiring only the limit from any fixed direction to be infinite.)
If the latter, this should be one of those funny counterexamples in analysis. On the other hand, the compactness of $S^1$ vaguely suggests to me that it has to be a pole.