TITLE: Cases where multiple induction steps are provably required
QUESTION [6 upvotes]: 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)

REPLY [6 votes]: 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.