Cantor's theorem for presheaves?
No such category exists. My original argument for this assumed local smallness and is below the break; here is a simpler argument that does not require local smallness (though it does basically use my original argument in the special case $\mathbf{C}=\mathbf{Set}$).
Let us take $\kappa$ to be an inaccessible cardinal and work with $V_\kappa$ as our universe, so the categories $\mathbf{Set}$ and $\mathbf{C}$ we start with are classes in $V_\kappa$ (so $\mathbf{Set}$ is the category of sets in $V_\kappa$), and we only go outside $V_\kappa$ to form functor categories. Suppose that there is an essentially surjective functor $\mathbf{C}\to\mathbf{Set}^{\mathbf{C}^{op}}$ and let $H$ be the associated functor $\mathbf{C}\times\mathbf{C}^{op}\to\mathbf{Set}$. I will obtain a contradiction by proving that there are $2^\kappa$ non-isomorphic objects of $\mathbf{Set}^{\mathbf{C}^{op}}$.
First of all, for every cardinal $\lambda<\kappa$, there is a constant presheaf $\lambda$ on $\mathbf{C}$, and so there is some object $A_\lambda$ with the property that $|H(A_\lambda,B)|=\lambda$ for all $B$. Now fix any object $B$, considered as an object of $\mathbf{C}^{op}$. Write $G(A)=H(A,B)$; then $G$ is a functor $\mathbf{C}\to\mathbf{Set}$ with the property that $|G(A_\lambda)|=\lambda$ for all $\lambda$. Consider $G$ as a functor $G^{op}:\mathbf{C}^{op}\to\mathbf{Set}^{op}$. We can then compose $G^{op}$ with any functor $P:\mathbf{Set}^{op}\to\mathbf{Set}$ to get a new presheaf $PG^{op}$ on $\mathbf{C}$. I claim that there are $2^\kappa$ choices of $P$ which give rise to non-isomorphic presheaves $PG^{op}$.
Indeed, since $G^{op}$ is essentially surjective, it suffices to give $2^\kappa$ different functors $P$ such that the induced maps $\{\text{cardinals }\lambda<\kappa\}\to\{\text{cardinals }\lambda<\kappa\}$ are distinct. This is not difficult; for instance, it can be done by a variant of the "wedge of spheres" construction below (let $\mathbf{C}=\mathbf{Set}$, $F(A)=|A|$, and instead of just taking a single copy of each sphere when constructing $T(Q)$, add enough spheres to change the cardinality of $T(Q)$ at $G(\alpha)$).
Let's work in the context of Grothendieck universes and require our categories to be locally small. Then I claim that no such category exists.
Let $\kappa$ be an inaccessible cardinal and let $V_\kappa$ be our base universe. Let $\mathbf{C}$ be a locally small category. Define an exhaustion of $\mathbf{C}$ to be an unbounded function $F:\operatorname{Ob}(\mathbf{C})\to \kappa$ such that if $B$ is a retract of $A$ then $F(B)\leq F(A)$.
First, I claim that if $\mathbf{C}$ has an exhaustion $F$, then $\mathbf{Set}^{\mathbf{C}^{op}}$ has $2^\kappa$ non-isomorphic objects and hence there is no essentially surjective functor $\mathbf{C}\to \mathbf{Set}^{\mathbf{C}^{op}}$. Let $1$ be the constant singleton presheaf on $\mathbf{C}$; let $*_B$ denote the unique element of $1(B)$ for all objects $B$. Given an object $A$ of $\mathbf{C}$, let $S^A$ (the "$A$-sphere", by analogy with the case $\mathbf{C}=\Delta$) be the presheaf obtained from $1$ by freely adjoining an element of $S^A(A)$ whose image under every map $A\leftarrow B$ is $*_B$ for all $B$ such that $F(B)<F(A)$. Since $A$ is not a retract of any such $B$, this new element of $S^A(A)$ will not be equal to $*_A$.
Now let $I\subseteq \kappa$ be the image of $F$ and choose a right inverse $G:I\to\operatorname{Ob}(\mathbf{C})$ of $F$. For each $Q\subset I$, define $T(Q)$ to be the colimit of the diagram consisting of the inclusions $1\to S^{G(\alpha)}$ for all $\alpha\in Q$. This colimit exists because for any object $A$, $1\to S^{G(\alpha)}$ is an isomorphism at $A$ for all $\alpha>F(A)$, and hence this colimit is small at $A$. We can determine the set $Q$ from the presheaf $T(Q)$ the same way you can determine the non-degenerate simplices of a simplicial set. Thus the presheaves $T(Q)$ are all non-isomorphic. Since there are $2^\kappa$ different values of $Q$, this proves the claim.
To prove the claimed theorem, it now suffices to show that any essentially large locally small category has an exhaustion. Let $\mathbf{C}$ be an essentially large locally small category, and assume WLOG it is skeletal. By essential largeness, let $f:\operatorname{Ob}(\mathbf{C})\to \kappa$ be a bijection. By local smallness, each object of $\mathbf{C}$ has fewer than $\kappa$ other objects as retracts (since a retraction is determined by the associated idempotent endomorphism). We can thus define $F:\operatorname{Ob}(\mathbf{C})\to \kappa$ by $$F(A)=\sup \{f(B):B\text{ is a retract of }A\},$$ and this $F$ will be an exhaustion.