As we know, if $\vec{F}=\nabla\times\vec{A}$ then from Stokes' Theorem, $\iint_{S_1} \vec{F}\dot \,d\vec{S}=\iint_{S_2}\vec{F}\dot \,d\vec{S}$ where $S_1$ and $S_2$ have the same boundary.
Does anyone have a quick example at the top of their head wherein the above equality is not satisfied implying that the vector potential of $\vec{F}$ does not exist?
Do you mean something like the inverse-square field $$ F(x, y, z) = \frac{(x, y, z)}{(x^{2} + y^{2} + z^{2})^{3/2}},\quad (x, y, z) \neq (0, 0, 0), $$ whose divergence vanishes identically, and $S$ the unit sphere, cut into any convenient pair of surfaces, e.g., the upper and lower hemispheres? (The integral of $F$ over $S$ with the outward unit normal is $4\pi$, not $0$.)