A classical example is $G$-$Set$ for a large group $G$. That this is a cocomplete elementary topos is not hard to see. Limits and colimits are formed at the underlying set level, and exponentials $Y^X$ are formed as usual as the set of functions $f: X \to Y$ with the $G$-action $(g, f) \mapsto g f$ defined by $g f: x \mapsto g f(g^{-1} x)$. The subobject classifier is the 2-element set with trivial $G$-action (again, as usual).
Thus $E = G$-$Set$ is a cocomplete (elementary) topos.
It remains to see $E$ is not Grothendieck. If it were, then every continuous functor such as the underlying-set functor $U: E \to Set$ would have a left adjoint $F$, by the special adjoint functor theorem (the hypotheses for the SAFT are satisfied: a Grothendieck $E$ is well-powered and complete and has a cogenerator $\Omega^c$ where $c$ is the coproduct of associated sheaves of objects in a small site presentation). In particular, we would have $\hom_E(F(1), -) \cong U$, but the only candidate for the representing object $F(1)$ would be $G$, which is ruled out by largeness.
Edit: Adam Epstein pointed out in a comment that the intuition in the last sentence, regarding the only candidate for the representing object, is not correct for some $G$ such as a large simple group. The following addendum patches up this oversight.
Let $G$ be a large free group, say the union of the diagram of free groups $F(\alpha)$ obtained by applying the free group functor to von Neumann cardinals $\alpha$ and initial segment inclusions between them. Then the topos $Set^G$ of $G$-sets is not Grothendieck.
Supposing it is, then the underlying-set functor $U: Set^G \to Set$ is representable, by the special adjoint functor theorem. Hence $U \cong \hom(X, -)$ for some $G$-set $X$. We will show $U$ has a proper class of non-isomorphic representable subfunctors, i.e., the object $X$ has a proper class of quotients, which is impossible in a Grothendieck topos (or even in a locally small topos -- quotients of $X$ are in bijection with equivalence relations on $X$, forming a subcollection of a hom-set $[X \times X, \Omega]$).
To begin with, for the class of canonical inclusions $i_\alpha: F(\alpha) \to G$ we may uniformly exhibit retractions $r_\alpha: G \to F(\alpha)$ (retractions in the sense of group homomorphisms). We may then view $F(\alpha)$ as a $G$-set with action $G \times F_\alpha \to F_\alpha$ taking $(g, x)$ to $r_\alpha(g) \cdot x$, and the representable functor $\hom(F(\alpha), -)$ is naturally a subfunctor of $U$. Indeed, a map $f: F(\alpha) \to A$ is uniquely determined by the value $a = f(1)$ at the identity element $1 \in F(\alpha)$, as the intertwiner condition yields $f(r_\alpha(g)) = g a$, whence $f(x) = i_\alpha(x)a$ for general $x \in F(\alpha)$. Denoting the idempotent map $i_\alpha r_\alpha: G \to G$ by $p_\alpha$, the subfunctor inclusion is given componentwise by
$$\hom(F(\alpha), A) \cong \{a \in A: \forall_{g \in G}\; g a = p_\alpha(g) a\} \hookrightarrow U(A)$$
and all these subfunctors are non-isomorphic, for the simple crude reason that the $F(\alpha)$ have generally different cardinalities as sets (as soon as $\alpha$ is in the uncountable range).
Best Answer
Hi. I mentioned that I had thought about this on nForum a while back - sorry I didn't get back to you sooner. The following sketch of a proof is mainly due to Colin McLarty.
Two features which distinguish a Grothendieck topos from a more general topos are
That it has a geometric morphism to Sets, namely the global sections functor.
That it has an object of generators (i.e. there is an object G such that if $f,g: A \to B$ are not equal then there exists an arrow $h: G \to A$ with $fh \neq gh$)
Let $C$ be a small complete category object in a Grothendieck topos $T$ which is not a preorder. Then $C^G$ is also a small complete category in this topos essentially because exponentials commute. The global sections functor applied to $C^G$ gives a small complete category in the category of sets which is not a preorder (the property of being a small complete category is preserved by geometric morphisms, and the special property of G allows the property of being "not a preorder" to carry through), which is a contradiction.
It is a little easier to think about in the case of sheaves on some topological space. There a small complete category object which is not a preorder would have to fail to be a preorder on some open set, and the sections on that open set would be a small complete category which is not a preorder. $G$ takes the place of this open set above.
If you have any questions about this let me know. In particular I can write out all of the adjunctions showing various properties are preserved, but I don't want to get too nitty gritty if it isn't useful to you.
Kind regards, Steven Gubkin