I thought about the theorem of Casorati-Weierstraß(in my lecture notes there are actually two, one deals with entire functions and the other with essential singularities) and asked myself if the following is true:
Let $G$ be a domain, $f(z)$ a holomorphic function on $G\setminus\{z_0\}$ and $z_0$ an essential singularity of $f(z)$.
$\Rightarrow f(z)$ is composed of a function $h(z) = g(a(z))$, where $g(z)$ is an entire transcendental function and $a(z)$ is a holomorphic function on $G\setminus\{z_0\}$ with a pole in $z_0$.