Is there a continuous closed curve $\gamma$ in $\mathbb{C}$ with the property that the mapping $$ \nu_{\gamma}: \mathbb{C} \backslash \operatorname{im}(\gamma) \rightarrow \mathbb{Z} $$ takes infinitely many values? Note that im is here the image, not the imaginary part and that $\nu_{\gamma}$ denotes the winding number $$ \nu_{\gamma}=\text{wind}_\gamma. $$
The closed curves, which one usually encounters in complex analysis do not seem to have this property.