Does there exits a smooth proper algebraic space $X$ over $\mathbb C$ with "large" fundamental group such that no finite etale cover of $X$ is a scheme?
By "large" fundamental group I mean that $X$ has infinitely many non-trivial finite etale covers (of increasing degree).
Equivalently, does there exist a Moishezon space $X$ with large fundamental group such that no finite degree topological covering of $X$ is the analytification of a scheme?
You can get one class of examples by modifying the Hironaka construction, cf. Appendix B, Examples 3.4.1 and 3.4.2 of Hartshorne's "Algebraic Geometry". What is perhaps surprising is that Hartshorne begins with a scheme -- the scheme from Example 3.4.1 -- that has a fixed point free $\mathbb{Z}/2\mathbb{Z}$-action, and then he forms the algebraic space as a quotient of this. So it appears that, by construction, there is a finite cover that is a scheme.
However, you can choose the ambient scheme so that the $\mathbb{Z}/2\mathbb{Z}$-action is fixed point free on a Zariski open that contains the relevant "exceptional surface", yet which has fixed points elsewhere. When you form the quotient and desingularize, the quotient will typically be simply connected. For instance, if the ambient scheme is a blowing up of projective space, then the quotient is rationally connected, hence any desingularization is simply connected. Finally, take the product of this simply connected algebraic space with any curve of genus $g\geq 1$.
