Toposes in which countable choice is true but dependent choice isn't I'd like examples of toposes in which Countable Choice is true but Dependent Choice isn't. I'd prefer examples without Excluded Middle. It's hard to find a natural example.
 A: Given any topological group $G$, the topos of sets with a continuous $G$-action is Boolean, and very often violates some choice principle or other. Under the translation between material and structural set theories, such toposes correspond to permutation (or Fraenkel–Mostowski) models of ZFA.
Theorem 8.12 in Jech's Axiom of Choice describes in terms in material sets a model of set theory in which countable choice (i.e. $\mathrm{AC}_{\aleph_0}$) holds, but DC doesn't. [In fact, Jech describes something more general, of which this is the "$<\aleph_1$ case"]
Consider the set $A := \aleph_1^{<\omega} = \bigcup_{n\in \omega} \aleph_1^n$ of finite sequences of countable ordinals. This carries a partial order where $s \leq t$ iff $t$ extends $s$, and is in fact a tree. Consider the automorphism group of this tree, call it $G$. This gets a topology by specifying a filter $F$ generated by an ideal $I\subset P(A)$. A subset $E\subset A$ is in $I$ precisely if it is a countable, bounded-height sub-tree. Then $F$ is the filter generated by the subgroups $\mathrm{Fix}(E) < G$ consisting of automorphisms that fix the subset $E \subset A$ pointwise.
Then the topos of continuous $G$-sets with this topological group is what you wanted.
