At the end of Remarque 2.3.6 (p. 221-222) of EGA I, the author says that there are functors in $\mathbf{Fais}|_{\mathbf{Ann}}$ (sheaf on the category of Rings) that are not isomorphic to sheaves that come from schemes. I would like to know one such example or if such example is constructed later on the book.
I'm adding the definition and context of each concept below:
A functor $G:\mathbf{Aff}^{op}\to\mathbf{Set}$ from the opposite category of affine schemes to the category of sets is called a presheaf. Given an affine scheme $X$, for any open subscheme $U$, one can consider the map $U\mapsto G(U)$. We say that $G$ is a sheaf when this map is always a sheaf in the usual sense.
Since there exist an equivalence of categories $F:\mathbf{Aff}^{op}\to\mathbf{Ring}$ between the category of affine schmes and the category of rings, that also defines an equivalence $\mathbf{Hom(Aff^{op},Set)}\cong\mathbf{Hom(Ring,Set)}$. Hece we can define a sheaf on the category of rings as a (covariant) functor $\mathbf{Ring}\to\mathbf{Set}$ whose image under the previous equivalence is a sheaf in the sense defined earlier.
Similarily we can define a sheaf on the category of schemes $\mathbf{Sch}$, but it turns out that the category of such sheaves is equivalent to that of sheaves on affine schemes. One can prove that, given an scheme $X$, the functor $h_X:Y\mapsto\mathrm{Hom}(Y,X)$ is a sheaf on $\mathbf{Sch}$, and since $h:X\mapsto h_X$ is fully faithful, we can identify the category of schemes with a subcategory of the sheaves on $\mathbf{Ring}$ by the previous equivalences.