User blog comment:B1mb0w/The R Function/@comment-25337554-20180305085245

That's just ω2. You got analysis completely wrong: Even though R(r(0),n) is f_ω(n), that does NOT mean R(R(r(0),r(0)),n) is f_{f_ω(ω)}(n). R(r(0),r(0)) is just a number (strictly speaking, it's a function), so it's just f_[some large number](n). And the nesting R^n(r(0)*,r(0)) finally reaches to ω+1.

If you don't believe, think about what comes next:

f_ω(f_ω(f_ω(f_ω(n))))

=f_ω(f_ω(f_ω(f_n(n))))

=f_ω(f_ω(f_{f_n(n)}(f_n(n))))

=[insert a formula here]

Note: Don't mention that "f_ω(ω)" does not exist. It's an intuitive one and it is supposed to mean φ(ω,0).