It's clear that the axiom of replacement can be used to construct very large sets, such as 
$$
\bigcup\_{i=0}^\infty P^i N,
$$
where $N$ is the natural numbers. I assume that it can be used to construct sets much lower in the Zermelo hierarchy, such as sets of natural numbers, but I don't know of an example. Is there an easy example? (Just to be clear, I mean an example that requires the use of replacement, not just one where you could use replacement if you wanted to.)


I would guess you can cook up an example using Borel determinacy, since that involves games of length $\omega$, but it would be great if there was an even more direct example.


Also, I'd be curious to know for any such examples at what stage they first come along in the constructible universe. $\omega + 1$? The first Church-Kleene ordinal? Some other ordinal I've never heard of?



Replacement and Sets of Natural Numbers
This probably isn't what you are looking for, but one can write down an explicit Diophantine equation for which ZFC proves that it has a solution, but ZFC minus replacement does not (assuming it is consistent). Namely, use Godel encoding and the solution of Hilbert's 10th problem to write down a Diophantine equation whose only solutions encode proofs of the consistency of "ZFC minus replacement." One wants a "naturally occurring" example instead, but it's hard to say what that means.


*(Edit: The following is corrected thanks to Andres' comments)*


For instance, I think the answer to Ricky's formulation in the comments is α = ω+1, but again probably not for the reason you expect. Namely, we can prove in ZFC that ZFC-Repl has a *countable* transitive model. To do this we start from an arbitrary transitive model (such as $V\_{\omega+\omega}$) and apply the downward Lowenheim-Skolem theorem to find a countable submodel. This countable submodel may no longer be transitive, but it is still well-founded, so by Mostowski's collapsing lemma it is isomorphic to an (also countable) transitive model.


Since ZFC-Repl has a countable transitive model, $V\_{\omega+1}$ (being uncountable) cannot be a subset of all such transitive models. But $V\_\omega$ is the set of hereditarily finite sets, which I think have to be in any transitive model since each of them can be uniquely characterized by a formula.


