Let $X$ be a smooth projective algebraic variety over $\mathbb{C}$. Then I think that someone (Serre?) showed that the Cohomological Etale Brauer Group agrees with the torsion part of the Analytic Brauer Group $H^{2}(X,\mathcal{O}^{\times})$. This latter group is calculated in the classical (metric) topology on the associated complex manifold with the sheaf of nowhere vanishing holomorphic functions.

However there can easily be non-torsion elements in $H^{2}(X,\mathcal{O}^{\times})$: for instance consider the image in $H^{3}(X,\mathbb{Z}) \cap (H^{(2,1)}(X) \oplus H^{(1,2)}(X))$.

Could there be a topology more refined than etale but defined algebraically which can see these non-torsion classes? Notice that one can also ask the question for any $H^{i}(X,\mathcal{O}^{\times})$. For $i=0,1$ the Zariski and etale work fine.

Why do things break down for $i>1$?