Let us consider the standard version of Stokes' Theorem, $$\iint_S\nabla\times\mathbf{F}\cdot d\mathbf{\sigma}=\oint_C\mathbf{F}\cdot d\mathbf{s}$$ where $S$ is a smooth orientable surface bounded by a connected, smooth simple closed curve $C$. Can one apply this to, say, the punctured torus? (A picture shown below.) The standard proof of Stokes' Theorem uses Green's theorem on the parametrization of the surface, so as long as we choose the parametrization to have a simply connected domain, Stokes' Theorem seems to hold.
