If a curve $C$ is a Koch curve (snowflake fractal) centered at origo, then is the following path integral defined? $$\int_C \frac{1}{z}\space dz.$$
If it exists, the value must be $2\pi i$ for positively oriented curve $C$. This might be a silly question, but I haven't find clear reference. So under what condition we can say that the winding number exists, in an integral sense, for a continuous curve?
The winding number is $2 \pi i$ as you say. With regard to whether that symbol you've written is "formally meaningful," Caratheodory's Theorem says that if you take a conformal isomorphism $\phi$ between the disc and the interior of a Jordan region, then this extends to a homeomorphism on the boundary.
In particular, it gives a parametrization of the boundary by $\phi(e^{i \theta}) = \phi(\gamma(\theta)), \theta \in [0, 2\pi)$ which you can then formally jam under a one-dimensional Riemann integral depending solely on the Riemann-integrability of Re$(\phi(e^{i \theta}))$ and Im$(\phi(e^{i \theta}))$. So instead of writing what you've written (which really isn't correct; I'd be shocked to hear that when you first defined a path integral, you didn't do so for piecewise-differentiable curves only), you should write $$\int_0^{2\pi} \frac{1}{\phi(\gamma(\theta))} \gamma'(\theta) d\theta$$
which does have a totally unambiguous meaning, assuming integrability. With regard to a comment made earlier, paths worse than $\mathcal{C}^1$ are plenty interesting - unless the Dirichlet Problem is uninteresting.