In accordance with the Great Picard theorem, the function $f(z) = e^{\frac 1 z}$ assumes every complex value except $0$ in every neighborhood of the origin.
I would like to know an elementary demonstration of that fact for teaching purposes.
I have already analyzed the behavior of $f$ along lines $t \mapsto (t x_0, t y_0)$ for some unit complex number $(x_0,y_0) \in \mathbb C$ as $t$ goes to zero. The values along those lines are $$ e^{ \dfrac{ x_0 }{t} } e^{ - \dfrac{ y_0 }{t} i } $$ It is clear that $f$ "spirals out of control" as $t$ vanishes. I am looking for an elementary proof that shows that every non-zero complex number is assumed by infinitely many parameter combinations $t > 0$ and unit $(x_0,y_0) \in \mathbb C$.
By using the bijection $z\mapsto \dfrac{1}{z}$, it should be clear this is equivalent to proving that outside of each ball $B(0,r)$ $e^z$ takes on every non-zero complex value infinitely many times.
Choose $\omega\in\mathbb{C}\space\backslash\{0\}$, and say $\alpha=\arg(\omega)$.
Then, for some $k_0\in\mathbb{Z}$ we have that $\alpha+2k_0\pi>r$.
Taking $z_0=\ln|\omega|+(\alpha+2k\pi) i$ for any $k\geq k_0$, we have that $z_0\notin B(0,r)$ and $e^{z_0}=\omega$, concluding the proof.