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?

I believe essential isolated singularities are preserved under non-constant, entire functions.
Indeed, if $g$ has an essential singularity at $0$, then Great Picard tells us that the image under $g$ of any punctured neighbourhood of $0$ is either $\mathbb C$ or $\mathbb C$ minus a point. Moreover, if $f$ is a non-constant entire function, then by applying Little Picard to $f$, we learn that the image under $f \circ g$ of any punctured neighbourhood of $0$ is $\mathbb C$ minus at most two points, and in particular, this image is dense in $\mathbb C$. Therefore, it is impossible for $f \circ g$ to have a pole or a removable singularity at $0$.
