Reference Request – Cases where it is proven that multiple introductory steps are required

I'm looking for references for theorems of the form:

1) Any proof of the theorem $ X $ requires $ n $ Applications of induction axioms

and specially

2) Any proof of the theorem $ X $ requires $ n $ nested applications of induction axioms.

I have seen similar propositions for the use of axioms other than induction axioms (and I would also be pleased with references, but I am particularly interested in induction)