According to a history of topos theory by McLarty, Blass (1989) showed that the existence of an object classifier over a given topos implies that the topos has a natural number object.
What is an object classifier, and how does it imply the existence of a natural number object?
Blass, A. (1989) Classifying topoi and the axiom of infinity. Algebra Universalis 26, no. 3, 341-345