This is an exercise from Conway's Functions of One Complex Variable(page 83, exercise 2).
Give an example of a closed rectifiable curve $\gamma$ in $\mathbb C$ s.t. for any integer $k$ there is a point $a\notin \{\gamma\}$ with $n(\gamma;a)=k$.
My solution:
$$ \gamma_1(t)=\begin{cases}\exp\{\frac{-1+i}{t}\} &t\in(0,1]\\0&t=0\end{cases}$$ is a rectifiable curve since $$\int_0^1|\gamma'(t)|\mathbb d t=\int_0^1\frac{\sqrt2}{t^2}e^{-1/t}\mathrm dt=\int_1^{\infty}{\sqrt2}{}xe^{-x}\mathrm dx=2\sqrt 2e^{-1}$$ If we connect $0$ and $\exp\{{-1+i}\}$ by a segment, then we obtain a desired closed rectifiable curve $\gamma$.
I can see the number of times $\gamma$ orbits about a given point intuitively, but how to compute the winding number rigorously? (Besides, Conway hasn't introduce homotopy there, so I don't know if there are other ways to evaluate the winding number except finding a curve homotopic to the original one?)