Here is what I know about the history of the cotangent complex:
Quillen did it over a point (i.e. for morphism of rings), Illusie did it in a topos (i.e. for sheaves of rings in a topos). And proved general theorems on deformation/obstructions. People have been using that to study various deformation problems of schemes. E.g. constructing obstruction theory. There are also generalization to 1-morphisms to stacks or to the logarithmic world. It seems to be a crucial object in the derived algebraic geometry setup.
My question is, has the cotangent complex used in other context with the full power of topoi? I understand that it is related to the solution of Serre's conjecture and the definition Andre-Quillen cohomology. I also understand to "glue" the affine construction is hard and you really need the power of topoi. I just want to know if this has been used in general on a morphism of rings in a topos. Or more generally, for a morphism of ringed topoi.
(I hope this question is not too vague, a more "concrete" example might be: does infinitesimal thickening of the first order of the fppf topos of a scheme $X$ (by a module in it) over that of a scheme $Y$ contains some interesting information?)
Just to be clear, by topos I always mean a Grothendieck topos, i.e. a category equivalent to the category of sheaves of sets over a site.