Cases where multiple induction steps are provably required I am looking for references for theorems of the form:
1) Any proof of theorem $X$ requires $n$ applications of induction axioms
and especially
2) Any proof of theorem $X$ requires $n$ nested applications of induction axioms.
I've seen similar statements for applications of axioms other than induction axioms (and I'd be happy to receive references for those, too, but I am particularly interested in induction)
 A: Here is an answer concerning recursion, rather than induction, but they are of course related. 
Namely, the Ackermann function is defined by a double nested recursion 
 $$A(m+1,n+1)=A(m,A(m+1,n))$$
with anchor cases defining $A(0,n)$ and $A(m,0)$. The function exhibits extremely rapid growth. 
I mention the function because one can prove that the Ackermann function is not a primitive recursive function, which are the functions one can construct from some primitive functions by closing under composition and simple recursion. 
Thus, the Ackermann function can be defined by a nested recursion, but not by a simple recursion.
A: Here is a reference for one way of making precise sense of your question and answering it:

Stefan Hetzl and Tin Lok Wong (2017): "Some observations on the logical foundations of inductive theorem proving", Logical Methods in Computer Science, Volume 13, Issue 4, doi:10.23638/LMCS-13(4:10)2017, arXiv:1704.01930

In Section 2.4 they show the following. Let $\text{PA}^-$ be the theory of the non-negative parts of discretely ordered rings (language: $\langle 0,1,+,\times,<\rangle$—see the paper for an axiomatization). Then for each theorem $\sigma$ of PA there is a formula $\varphi(x)$ such that $\text{PA}^-$ proves $\varphi(0)$ and $\forall x(\varphi(x)\implies\varphi(x+1))$ and $\forall x.\varphi(x)\implies\sigma$. Thus in this sense one application of induction always suffices.
