User blog comment:Emlightened/Intro/@comment-1605058-20160109180510/@comment-27513631-20160109205041

There exists a function F such that all of the functions in the definition of F, except F itself, are primitive recursive. For instance, we can define \(f_\omega\) this way using the infinite family of PR functions \(f_n\), but we cannot define \(f_{\omega+1}\) or \(f_{\omega2} using this method, as they would both reference the function \(f_\omega\), which isn't primitive recursive.

Consider the family of PR functions \(q_i\) defined as \(q_i(n)=(n/p_i)\cdot (p_{i-1})^{n}\), and the function \(Q\), such that \(Q(k,n)=Q(Q(k+1,q_i(n)),q_i(n))\) where \(i\) is the index of the least odd prime factor of \(n\), and \(Q(k,2^n)=k+n\) if no such factor exists. \(Q\) is a middle-growing hierarchy, which represents its ordinal through transforming \(n=2^{k_1}3^{k_2}5^{k_3}\cdots\) into \(\cdots\omega^2k_3+\omega k_2+k_1\). It's not hard to show that \(G(n)=Q(p_n,n)\) has a growth rate of \(\omega^\omega\) on the middle-growing hierarchy and the fast-growing hierarchy, and that each \(q_i\) is PR. I was asserting that a similar method exists that allows someone to reach \(\theta(\Omega^\omega\omega)\), although this is obviously much more complicated.