TITLE: Example of a topos that violates countable choice
QUESTION [6 upvotes]: 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?

REPLY [7 votes]: 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"?