Are essential isolated singularities preserved under non-zero holomorphic functions? 
Question. In univariate complex analysis, are essential isolated singularities preserved under non-zero holomorphic functions?

For example, if we've already proved that $e^{1/z}$ has an essential singularity at $0$, can we deduce that $e^{e^{1/z}}$ also has one at zero, without making any computations?

Let $f\colon D\longrightarrow\mathbb C$ be an analytic function and suppose that it has an essential singularity at some point $z_0$. Let $g$ be a non-constant entire function. Then $g\circ f$ also has an essential singularity at $z_0$. This is so because, by the Casorati-Weierstrass, if $U$ is a neighborhood of $z_0$ such that $U\setminus\{z_0\}\subset D$, then $f\bigl(U\setminus\{z_0\}\bigr)$ is a dense subset of $\mathbb C$. On the other and, it is an easy corollary of the Liouville theorem that the image of $g$ is dense. Therefore, $(g\circ f)(D)$ is dense too. And it follows from this that $g\circ f$ also has an essential singularity at $z_0$.
