The fast growing hierarchy "FGH" is a very explosive family of function, you create it using this simple sequence: f_0(n)=n+1
f_1(n)=f_0(n) iterated n times=2n
f_2(n)=f_1(n) iterated n times=(2^n)×n
and you get the point
For limit ordinals such as omega: f_w(n) or f_{w[n]}(n) equalates to f_n(n) meaning f_w eventually dominates any fixed finite k on f_k,
Rule in FGH:"f_a+1(n)=f_a(n) iterated n times" meaning f_w+1(n)=f_w iterated n times, this creates a chain reaction of fast growing functions
For refrence:grahams function is bounded by f_w+1(n), f_w+2(n) dominates Grahams function by not a tiny bit but by a large large large margin.
For fixed points and extended:
f_{w×2}(n) or f_{w+w}(n) equals f_{w+n}(n).
f_{(w×2)+1}(n)=f_{w×2} iterated n times, meaning f_{w×k}=f_{w×(k-1)+n}(n)
f_{w×w}=f_{w×n}
f_{(w×w)×2}=f_{(w×w)+n}
f_{(w×w)×3}(n)=f_{((w×w)×2)+(w×n)}
f_{w^3}=f_{((w×w)×n-1)+(w×n)}.