User blog:AJZajac/How fast does this sequence grow (problem in ordinal arithmetic)?

What would be the growth rate for a function a(n) defined as:

a(0) = 1

a(n+1) = f ω +a(n)(n)

With f representing a function in the FGH



So that:

.a(1) = fω(1) = 2

.a(2) = fω2(2)

.a(3) = fωf_ω2(2)(3)  etc.

I know it would be easy to diagonalize if the recurvsive formula was a(n+1) = f ω +a(n) (a(n)), but how much slower does the sequence grow because using n instead of a(n) ?