As I understand, one of the reasons for "bootstrapping" to the category of algebraic spaces before constructing the category of Artin stacks is that algebraic spaces form a stack in the etale (at least) topology, while schemes do not, even though one frequently has (at least in other contexts) to use the fact that, say, affine or quasi-affine schemes do form a stack even in the fpqc topology (by descent theory for quasi-coherent sheaves). As a result, I'm curious: what is the simplest example of non-gluable (say, etale) descent data for schemes?

To clarify, I'm looking for an example of an fpqc morphism $Y' \to Y$, a scheme $X' \to Y'$ together with the usual patching after pull-back to $Y' \times_Y Y'$ that does not come from a scheme over $Y$.

on some (affine) scheme. If you replace Sch (or Aff) by the topos with the canonical topology, then you're correct since $F$ only needs to be a scheme etale locally on itself. $\endgroup$ – Anton Geraschenko Aug 5 '11 at 12:39