In any topos, the Lawvere-Tierney topology given by $\neg\neg:\Omega\to\Omega$ is the strongest for which the morphism $0\to 1$ is closed. In most of the examples I am familiar with (admittedly, these are mostly presheaf categories on finite diagrams), the only topology above $\neg\neg$ is actually the inconsistent topology; but since no one has ever said otherwise, I'm guessing there are examples of toposes with topologies strictly between double negation and inconsistency. I haven't yet been able to come up with an example, though.
Question: Are there any easy examples of a topos $\mathcal{E}$ (preferably a presheaf category) and a topology $j:\Omega\to\Omega$ in $\mathcal{E}$ that lies strictly between $\neg\neg$ and inconsistency?
Certainly.
Let $B$ be a Boolean nontrivial topos [perhaps the category of sets if you're using classical metalogic], and consider the topos $B^2$ which has subobject classifier $(\Omega, \Omega)$.
Let $x = (\top, \bot) : (1, 1) \to (\Omega, \Omega)$.
Define $j(u) = x \lor u$. Then $j$ is a Lawvere-Tierney topology, since $j(\top) = x \lor \top = \top$, $j(a \land b) = x \lor (a \land b) = (x \lor a) \land (x \lor b) = j(a) \land j(b)$, and $j(j(u)) = x \lor (x \lor u) = (x \lor x) \lor u = x \lor u = j(u)$.
Furthermore, it is stronger than the $\neg \neg$ topology (which is just the identity map). And it is weaker than the inconsistent topology, since $j(\bot) = x \neq \top$.