Can the winding number be infinite? Let $z$ be a point in the complex plane, and $\gamma$ be a closed curve. Is it possible that 
$$n(\gamma,z) =  \frac{1}{2\pi i}\int_\gamma \frac{dw}{w-z}$$
becomes unbounded? In other words, is it possible to find a curve $\gamma$ such that it winds around a fixed point infinitely many times?

Without loss of generality, we show that the winding number around $0$ is finite (the general proof follows by translating the complex plane). 
Remember the alternative definition of the winding number: if $\gamma$ is a closed path, we can choose a continuous choice of argument $\theta:[0,1] \to \mathbb R$ such that $$\frac{\gamma(t)}{|\gamma(t)|}=\mathrm e^{i\theta(t)}$$
The winding number can then be defined as$$w(\gamma, 0)=\frac{1}{2\pi}\big(\theta(1) - \theta(0)\big)$$
But $\theta$ is a continuous function on a compact set so it is bounded. In particular the winding number is finite.
