When $C$ is essentially small, the presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is the free cocompletion of $C$. The presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is also a topos.

When $C$ is truly large, the presheaf category $[C^\mathrm{op},\mathsf{Set}$] is not even locally small. But the full subcategory $\mathcal{P}C \subset [C^\mathrm{op},\mathsf{Set}]$, consisting of the presheaves which are small colimits of representables, is still the free cocompletion of $C$. When $C$ is small, $\mathcal{P}C$ is the whole category $[C^\mathrm{op},\mathsf{Set}]$, so it is a topos. But when $\mathcal{C}$ is large, is $\mathcal{P}C$ a topos?

Day and Lack, following Rosicky, study when $\mathcal{P}C$ is complete, and when it is cartesian closed. Certainly cartesian closedness is necessary to be a topos; I believe that a cocomplete topos is also complete, so that completeness is also necessary. But I suspect that $\mathcal{P}C$ might only have a subobject classifier if $\mathcal{C}$ is essentially small.

If it makes a difference to restrict to sheaves in some Grothendieck topology, that would be interesting to know. I suppose my motivating example is when $C$ is the category of schemes (or just affine schemes). Algebraic geometers are very interested in sheaves on this category in various topologies; I suspect most such interesting sheaves are small, but I don't know if that restriction gives one a topos to work with.

**EDIT**

As David Carchedi points out in the comments, I should emphasize that I'm interested in when $\mathcal{P}C$ is an *elementary* topos (a finitely complete, cartesian closed category with a subobject classifier). As Eric Wofsey argues in the comments, if $C$ is essentially large, then $\mathcal{P}C$ is never a Grothendieck topos (it doesn't have a small generating set).

**EDIT**

The weakest of observations: the Yoneda embedding $C \to \mathcal{P}C$ is still continuous, so preserves monos, and is fully faithful; so it induces injections on subobject lattices. So if $C$ is not well-powered to begin with, then $\mathcal{P}C$ is also not well-powered, so can't be a topos (given that it's locally small: look at the size of $\mathrm{Hom}(a,\Omega)$).

**EDIT**

It looks like the canonical way to give a general answer to this problem would be to generalize the work of Menni which characterizes when the exact completion of a finitely complete category is a topos. Taking the category of small sheaves is put in the same framework as exact completions in Mike Shulman's paper which Adeel linked to in the comments. So the way forward ought to be clear. I'm not sure how easy criteria along Menni's lines would be to check in particular cases like schemes, though.