Given a Grothendieck topology $T$, say subcanonical, on a category $C$, we are able to talk about sheaves in $(C, T)$. Since pre/sheaves can be viewed as generalized objects of $C$ (via Yoneda functor), one natural question arises, is $Shv(C, T)$ or $Psh(C)$ a site extending the site $(C, T)$? If it is, then we take pre/sheaves on that new site, can we get more pre/sheaves?
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Yes. Every topos has the so-called canonical topology, in which a sieve is covering if and only if it is generated by a small jointly epimorphic family. (In the Grothendieck topos case, this is the same as being jointly epimorphic, because the small-generation condition is automatic.) Moreover, the category of sheaves in the canonical topology of a Grothendieck topos is equivalent to the original topos, so there are no new sheaves. See Proposition 2.2.7 in [Johnstone, Sketches of an elephant, Part C].
That said, it is not legitimate to form presheaves on a large site: for one thing, the resulting category is not a topos.