User blog comment:Mh314159/FOX notation/@comment-35470197-20191204015153/@comment-35470197-20191205011401

One of the simplest way is to put the following rules to compute f(x) for any sequence S of non-negative integers and non-negative integer x: Then the limit is again ω^ω. For example, we have and so on. This is precisely how FGH associated to an ordinal notation up to ω^ω works except for the minour change using the subscript.
 * 1) If S = (0), then f(x) = x+1.
 * 2) If S = (n+1,X) for a non-negative integer n and a possibly empty sequence X, then f(x) = f<(n,X)>_x(x).
 * 3) If S = (Z,0,n+1,X) for a possibly empty sequence Z of zeros, a non-negative integer n, and a possibly empty sequence X, then f(x) = f<(Z,x,n,X)>(x).
 * 1) f<(0,0,c+1)>(x) = f<(0,x,c)>(x)
 * 2) f<(1,0,c+1)>(x) = f<(0,0,c)>_x(x)
 * 3) f<(0,1,c+1)>(x) = f<(x,0,c)>(x)