TITLE: Are non-algebraic stacks useful in algebraic geometry?
QUESTION [13 upvotes]: The title is a bit vague. What I want to know is if there is any geometric application of non-algebraic stacks. I know e.g. the category of coherent sheaves is an example. But I want to ask if people think about some of the general stacks geometrically (instead of merely thinking about them as fibered categories). 
If the paragraph still doesn't explain my purpose, let me be more specific: people talk about "locally around a curve in M_g" something happens. But they don't say "locally near a coherent sheaf in Coh", right?
I read Vistoli's notes on general stacks before I saw people talk about algebraic stacks geometrically, and I always feel like for me there is a gap between these two ways of thinking. I feel like e.g. when a birational geometer uses stacks as a tool he probably don't put general fibered categories in his mind, (although abstract nonsense shouldn't be a big problem for him) right?
[A related question is, is category over Schemes fibered in something other than groupoid useful in geometry? Why people only care about categories fibered in groupoids?]

REPLY [9 votes]: The answer is a qualified yes, and it depends on how broadly you define "algebraic geometry".  Here is a class of examples:
A sheaf with flat connection on a smooth variety is equivalent to a sheaf on the de Rham stack of that variety, and de Rham stacks of positive dimensional varieties are not algebraic.  You are basically taking a quotient with respect to the action of an infinitesimal symmetry, and the corresponding covering morphism satisfies only a formal version of smoothness.  More generally, there are variations on the notion of sheaf with connection, e.g., action of an algebroid, that can be viewed functorially using sheaves on non-algebraic stacks which are stackifications of reasonably nice formal groupoids.  
There is a sketch of a theory of such stacks in sections 2.9.10-2.9.12 of Chiral Algebras by Beilinson and Drinfeld, but in fact, the entire book is secretly about them, restricted to the language of $D$-modules.   This is a situation where you can do some kind of "geometry", but it is a more degenerate environment than most algebraic geometers tend to consider.  For example, you may end up using "compound tensor structures" because ordinary tensor products on quasicoherent sheaves, with their nice properties, are unavailable.