= Solution
At every $z_k$, the <function> $\tan(1/z)$ has a simple pole. The <exponential of a pole is an essential singularity>, so
$$
\boxed{z_k=\frac{2}{(2k+1)\pi}\quad(k\in\mathbb Z)}
$$
are essential singularities of $\exp(\tan(1/z))$.
These points accumulate at $0$. Therefore no punctured neighbourhood of zero is a domain of holomorphy for the <function>: zero is a <non-isolated singularity>, specifically an accumulation point of essential singularities, rather than an isolated essential singularity. There are no other singularities in $\mathbb C$.
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