Example of a topos that violates countable choice At this nLab page we have the line

In contrast, any topos that violates countable choice, of which there are plenty, must also violate internal COSHEP.

It doesn't give an example, and neither does the page on countable choice. So, what are these all-so-common examples?
 A: Sheaves on the unit interval also break countable choice (and also weak countable choice, where you only require at most one of the sets to not be a singleton, which is weaker than both CC and LEM).
In particular, the Dedekind reals and the Cauchy reals fail to be the same in that topos (the former are continuous Real functions on the interval, the latter are just the reals from the category of sets), while they are equal in any topos where Countable Choice or LEM holds.
A: If you're looking for a purely topos-theoretic model, I think you don't need to go through set theory (even though the end result may end up being basically equivalent).  Look at the topos of continuous actions of the pro-completion of the integers, which is to say, the category of sets equipped with an automorphism all of whose orbits are finite.  Here we have an N-indexed family of objects (one orbit of each cardinality) which are all inhabited, but whose product is empty -- hence the NNO is not internally projective.
Have you read P. Freyd's paper "The Axiom of Choice"?
A: One sort of examples consists of the topoi of sets and functions obtained from models of ZF that violate countable choice.  The original Cohen model is among these, and so are many others.  Perhaps easier to understand are permutation models of ZFA (the variant of ZF that allows for atoms (= urelements)).  The basic Fraenkel model, the second Fraenkel model, and Mostowski's linearly ordered model (probably the three best-known permutation models --- see Chapter 4 of Jech's book "The Axiom of Choice") all have infinite Dedekind-finite sets and therefore violate countable choice.
