Finding a curve with a condition on winding numbers

Question

I want to find a continuous and closed curve $\gamma$ so that the map $\nu_{\gamma}:\mathbb{C}$\Im$(\gamma)\to \mathbb{Z}$

takes infintely many values. Here $\nu_{\gamma}(a)$ is the winding number of $\gamma$ around $a$.

This map exists (by a remark in a textbook). I played around by expressions containing $\sin(1/t)$ to get something which oscillates a lot but can't conclude.

Answer 1

One possible answer is $\gamma(t):=t\cdot(t-1)e^{\frac{i}{t}}$ (with $\gamma(0):=0$) for $t\in[0,1]$.