I propose that the limit of Fa#a#a#...#a#a#a (with b a's) do not reach Fa##b like in the Hyper-E notation, but to reach Fa[1+#]b instead.
Let @ to be rest of the expression that remains unchanged when one of the rules is applied.
The integer preceding the entry a(n) is represented by z(n).
We reveal the first few rules of the notation:
If there are no hyperions:
F[b]x = b^x.
If the last entry is 1:
F[b]@[z(n-1)+#]a(n)[z(n)+#]1 = F[b]@[z(n-1)+#]a(n)
If n = 0:
F[b]@[0+#]a(n) = F[b]@#a(n)
If n > 0:
F[b]@[z(n-2)+#]a(n-1)[z(n-1)+#]a(n) = F[b]@[z(n-2)+#]a(n-1)[(z(n-1)-1)+#]a(n-1)[z(n-1)+#](a(n) - 1)
Otherwise:
F[b]@[z(n-2)+#]a(n-1)#a(n) = F[b]@[z(n-2)+#](F[b]@[z(n-2)+#]a(n-1)#(a(n) - 1)) (note [0+#] = #)
Here is the examples:
F100 = 10^100
F[2]100 = 2^100
F100#2 = F(F100#1) = FF100 = F(10^100) = 10^10^100
F100#100 = F(F100#99) = FF(F100#98) = ... = FFF...FFF100 (100 F's)
F100#100#100 = F100#(F100#100#99) = ...
F100[1+#]100 = F100#100[1+#]99 = F100#100#100[1+#]98 = ... = F100#100#100#...#100#100#100 (100 100's)
F100[1+#]100[1+#]4 = F100[1+#]100#100#100#100
F100[2+#]100 = F100[1+#]100[1+#]...[1+#]100[1+#]100 (100 100's)
F100[3+#]100 = F100[3+#]100[3+#]...[3+#]100[3+#]100 (100 100's)
Then...
F100[x+#]100 = F100[#+1]x?
Let's develop further!
Let's analyze the ordinal level!
F[b]a[1+#]a ~ ordinal level ω
F[b]a[1+#]a#a ~ ordinal level ω + 1
F[b]a[1+#]a#a#a ~ ordinal level ω + 2
F[b]a[1+#]a#a#a#a ~ ordinal level ω + 3
F[b]a[1+#]a[1+#]a ~ ordinal level ω2
F[b]a[1+#]a[1+#]a#a ~ ordinal level ω2 + 1
F[b]a[1+#]a[1+#]a[1+#]a ~ ordinal level ω3
F[b]a[1+#]a[1+#]a[1+#]a[1+#]a ~ ordinal level ω4
F[b]a[2+#]a ~ ordinal level ω^2
F[b]a[2+#]a#a ~ ordinal level ω^2 + 1
F[b]a[2+#]a[1+#]a ~ ordinal level ω^2 + ω
F[b]a[2+#]a[2+#]a ~ ordinal level ω^2*2
F[b]a[2+#]a[2+#]a[2+#]a ~ ordinal level ω^2*3
F[b]a[3+#]a ~ ordinal level ω^3
F[b]a[3+#]a[3+#]a ~ ordinal level ω^3*2
F[b]a[4+#]a ~ ordinal level ω^4
F[b]a[5+#]a ~ ordinal level ω^5
F[b]a[6+#]a ~ ordinal level ω^6
This is the intended ordinal level of the ill-defined level (?)
F[b]a[#+1]a ~ ordinal level ω^ω
F[b]a[#+1]a#a ~ ordinal level ω^ω + 1
F[b]a[#+1]a[1+#]a ~ ordinal level ω^ω + ω
F[b]a[#+1]a[2+#]a ~ ordinal level ω^ω + ω^2
F[b]a[#+1]a[3+#]a ~ ordinal level ω^ω + ω^3
F[b]a[#+1]a[#+1]a ~ ordinal level ω^ω*2
F[b]a[#+1]a[#+1]a[#+1]a ~ ordinal level ω^ω*3
F[b]a[1+[#+1]]a ~ ordinal level ω^(ω + 1)
F[b]a[1+[#+1]]a[1+[#+1]]a ~ ordinal level ω^(ω + 1)*2
F[b]a[2+[#+1]]a ~ ordinal level ω^(ω + 2)
F[b]a[3+[#+1]]a ~ ordinal level ω^(ω + 3)
F[b]a[4+[#+1]]a ~ ordinal level ω^(ω + 4)
F[b]a[#+[#+1]]a ~ ordinal level ω^ω2
F[b]a[1+[#]+[#+1]]a ~ ordinal level ω^(ω2 + 1)
F[b]a[2+[#]+[#+1]]a ~ ordinal level ω^(ω2 + 2)
F[b]a[[1+#]+[#+1]]a ~ ordinal level ω^ω3
F[b]a[1+[1+#]+[#+1]]a ~ ordinal level ω^(ω3 + 1)
F[b]a[[2+#]+[#+1]]a ~ ordinal level ω^ω4
F[b]a[[3+#]+[#+1]]a ~ ordinal level ω^ω5
F[b]a[1+#+1]a ~ ordinal level ω^ω^2
F[b]a[1+[1+#+1]]a ~ ordinal level ω^(ω^2 + 1)
F[b]a[2+[1+#+1]]a ~ ordinal level ω^(ω^2 + 2)
F[b]a[#+[1+#+1]]a ~ ordinal level ω^(ω^2 + ω)
F[b]a[1+[#]+[1+#+1]]a ~ ordinal level ω^(ω^2 + ω + 1)
F[b]a[[1+#]+[1+#+1]]a ~ ordinal level ω^(ω^2 + ω2)
F[b]a[[2+#]+[1+#+1]]a ~ ordinal level ω^(ω^2 + ω3)
F[b]a[[#+1]+[1+#+1]]a ~ ordinal level ω^(ω^2*2)
F[b]a[1+[#+1]+[1+#+1]]a ~ ordinal level ω^(ω^2*2 + 1)
F[b]a[#+[#+1]+[1+#+1]]a ~ ordinal level ω^(ω^2*2 + ω)
F[b]a[[1+[#+1]]+[1+#+1]]a ~ ordinal level ω^(ω^2*3)
F[b]a[[2+[#+1]]+[1+#+1]]a ~ ordinal level ω^(ω^2*4)
F[b]a[2+#+1]a ~ ordinal level ω^ω^3
F[b]a[1+[2+#+1]]a ~ ordinal level ω^(ω^3 + 1)
F[b]a[#+[2+#+1]]a ~ ordinal level ω^(ω^3 + ω)
F[b]a[[#+1]+[2+#+1]]a ~ ordinal level ω^(ω^3 + ω^2)
F[b]a[[1+#+1]+[2+#+1]]a ~ ordinal level ω^(ω^3*2)
F[b]a[[1+#+1]+[2+#+1]]a ~ ordinal level ω^(ω^3*2)
?????
Coming very soon...
From ω^ω^2-level:
F[b]a[#+2]a ~ ordinal level ω^ω^2
F[b]a[1+[#+2]]a ~ ordinal level ω^(ω^2 + 1)
F[b]a[2+[#+2]]a ~ ordinal level ω^(ω^2 + 2)
F[b]a[[#]+[#+2]]a ~ ordinal level ω^(ω^2 + ω)
F[b]a[1+[#]+[#+2]]a ~ ordinal level ω^(ω^2 + ω + 1)
F[b]a[[1+#]+[#+2]]a ~ ordinal level ω^(ω^2 + ω2)
F[b]a[[2+#]+[#+2]]a ~ ordinal level ω^(ω^2 + ω3)
F[b]a[[#+1]+[#+2]]a ~ ordinal level ω^(ω^2*2)
F[b]a[1+[#+1]+[#+2]]a ~ ordinal level ω^(ω^2*2 + 1)
F[b]a[[#]+[#+1]+[#+2]]a ~ ordinal level ω^(ω^2*2 + ω)
F[b]a[[1+#]+[#+1]+[#+2]]a ~ ordinal level ω^(ω^2*2 + ω2)
F[b]a[[1+#+1]+[#+2]]a ~ ordinal level ω^(ω^2*3)
F[b]a[[2+#+1]+[#+2]]a ~ ordinal level ω^(ω^2*4)
F[b]a[#+3]a ~ ordinal level ω^ω^3
F[b]a[1+[#+3]]a ~ ordinal level ω^(ω^3 + 1)
F[b]a[[#]+[#+3]]a ~ ordinal level ω^(ω^3 + ω)
F[b]a[[1+#]+[#+3]]a ~ ordinal level ω^(ω^3 + ω2)
F[b]a[[#+1]+[#+3]]a ~ ordinal level ω^(ω^3 + ω^2)
F[b]a[[1+#+1]+[#+3]]a ~ ordinal level ω^(ω^3 + ω^2*2)
F[b]a[[#+2]+[#+3]]a ~ ordinal level ω^(ω^3*2)
F[b]a[[1+#+2]+[#+3]]a ~ ordinal level ω^(ω^3*3)
F[b]a[#+4]a ~ ordinal level ω^ω^4
F[b]a[[#+3]+[#+4]]a ~ ordinal level ω^(ω^4*2)
F[b]a[#+5]a ~ ordinal level ω^ω^5
F[b]a[#+6]a ~ ordinal level ω^ω^6
From ω^ω^ω-level:
F[b]a[#+#]a ~ ordinal level ω^ω^ω
F[b]a[#+#]a[#+#]a ~ ordinal level ω^ω^ω*2
F[b]a[1+[#+#]]a ~ ordinal level ω^(ω^ω + 1)
F[b]a[2+[#+#]]a ~ ordinal level ω^(ω^ω + 2)
F[b]a[[#]+[#+#]]a ~ ordinal level ω^(ω^ω + ω)
F[b]a[[1+#]+[#+#]]a ~ ordinal level ω^(ω^ω + ω2)
F[b]a[[#+1]+[#+#]]a ~ ordinal level ω^(ω^ω + ω^2)
F[b]a[[#+2]+[#+#]]a ~ ordinal level ω^(ω^ω + ω^3)
F[b]a[[#+3]+[#+#]]a ~ ordinal level ω^(ω^ω + ω^4)
F[b]a[1+#+#]a ~ ordinal level ω^(ω^ω*2)
F[b]a[2+#+#]a ~ ordinal level ω^(ω^ω*3)
F[b]a[3+#+#]a ~ ordinal level ω^(ω^ω*4)
F[b]a[#+1+#]a ~ ordinal level ω^ω^(ω + 1)
F[b]a[1+[#+1+#]]a ~ ordinal level ω^(ω^(ω + 1) + 1)
F[b]a[[#]+[#+1+#]]a ~ ordinal level ω^(ω^(ω + 1) + ω)
F[b]a[[#+1]+[#+1+#]]a ~ ordinal level ω^(ω^(ω + 1) + ω^2)
F[b]a[[#+#]+[#+1+#]]a ~ ordinal level ω^(ω^(ω + 1) + ω^ω)
F[b]a[1+[#+#]+[#+1+#]]a ~ ordinal level ω^(ω^(ω + 1) + ω^ω + 1)
F[b]a[[#]+[#+#]+[#+1+#]]a ~ ordinal level ω^(ω^(ω + 1) + ω^ω + ω)
F[b]a[[1+#+#]+[#+1+#]]a ~ ordinal level ω^(ω^(ω + 1) + ω^ω*2)
F[b]a[[2+#+#]+[#+1+#]]a ~ ordinal level ω^(ω^(ω + 1) + ω^ω*3)
F[b]a[1+#+1+#]a ~ ordinal level ω^(ω^(ω + 1)*2)
F[b]a[2+#+1+#]a ~ ordinal level ω^(ω^(ω + 1)*3)
F[b]a[#+2+#]a ~ ordinal level ω^ω^(ω + 2)
F[b]a[1+[#+2+#]]a ~ ordinal level ω^(ω^(ω + 2) + 1)
F[b]a[[#]+[#+2+#]]a ~ ordinal level ω^(ω^(ω + 2) + ω)
F[b]a[[#+#]+[#+2+#]]a ~ ordinal level ω^(ω^(ω + 2) + ω^ω)
F[b]a[[1+#+#]+[#+2+#]]a ~ ordinal level ω^(ω^(ω + 2) + ω^ω*2)
F[b]a[[#+1+#]+[#+2+#]]a ~ ordinal level ω^(ω^(ω + 2) + ω^(ω + 1))
F[b]a[[1+#+1+#]+[#+2+#]]a ~ ordinal level ω^(ω^(ω + 2) + ω^(ω + 1)*2)
F[b]a[1+#+2+#]a ~ ordinal level ω^(ω^(ω + 2)*2)
F[b]a[#+3+#]a ~ ordinal level ω^ω^(ω + 3)
F[b]a[#+4+#]a ~ ordinal level ω^ω^(ω + 4)
F[b]a[#+[#]+#]a ~ ordinal level ω^ω^ω2
F[b]a[1+#+[#]+#]a ~ ordinal level ω^(ω^ω2*2)
F[b]a[2+#+[#]+#]a ~ ordinal level ω^(ω^ω2*3)
F[b]a[#+1+[#]+#]a ~ ordinal level ω^ω^(ω2 + 1)
F[b]a[#+2+[#]+#]a ~ ordinal level ω^ω^(ω2 + 2)
F[b]a[#+[1+#]+#]a ~ ordinal level ω^ω^ω3
F[b]a[#+[2+#]+#]a ~ ordinal level ω^ω^ω4
F[b]a[#+[3+#]+#]a ~ ordinal level ω^ω^ω5
F[b]a[#+#+1]a ~ ordinal level ω^ω^ω^2
F[b]a[1+[#+#+1]]a ~ ordinal level ω^(ω^ω^2 + 1)
F[b]a[[#]+[#+#+1]]a ~ ordinal level ω^(ω^ω^2 + ω)
F[b]a[[#+#]+[#+#+1]]a ~ ordinal level ω^(ω^ω^2 + ω^ω)
F[b]a[1+#+#+1]a ~ ordinal level ω^(ω^ω^2*2)
F[b]a[#+1+#+1]a ~ ordinal level ω^ω^(ω^2 + 1)
F[b]a[#+2+#+1]a ~ ordinal level ω^ω^(ω^2 + 2)
F[b]a[#+[#]+#+1]a ~ ordinal level ω^ω^(ω^2 + ω)
F[b]a[#+1+[#]+#+1]a ~ ordinal level ω^ω^(ω^2 + ω + 1)
F[b]a[#+[1+#]+#+1]a ~ ordinal level ω^ω^(ω^2 + ω2)
F[b]a[#+[2+#]+#+1]a ~ ordinal level ω^ω^(ω^2 + ω3)
F[b]a[#+[#+1]+#+1]a ~ ordinal level ω^ω^(ω^2*2)
F[b]a[#+[1+#+1]+#+1]a ~ ordinal level ω^ω^(ω^2*3)
F[b]a[#+[2+#+1]+#+1]a ~ ordinal level ω^ω^(ω^2*4)
F[b]a[#+#+2]a ~ ordinal level ω^ω^ω^3
F[b]a[#+1+#+2]a ~ ordinal level ω^ω^(ω^3 + 1)
F[b]a[#+[#]+#+2]a ~ ordinal level ω^ω^(ω^3 + ω)
F[b]a[#+[#+1]+#+2]a ~ ordinal level ω^ω^(ω^3 + ω^2)
F[b]a[#+[1+#+1]+#+2]a ~ ordinal level ω^ω^(ω^3 + ω^2*2)
F[b]a[#+[#+2]+#+2]a ~ ordinal level ω^ω^(ω^3*2)
F[b]a[#+[1+#+2]+#+2]a ~ ordinal level ω^ω^(ω^3*3)
F[b]a[#+#+3]a ~ ordinal level ω^ω^ω^4
F[b]a[#+#+4]a ~ ordinal level ω^ω^ω^5
From ω^ω^ω^ω-level:
F[b]a[#+#+#]a ~ ordinal level ω^ω^ω^ω
F[b]a[1+[#+#+#]]a ~ ordinal level ω^(ω^ω^ω + 1)
F[b]a[1+#+#+#]a ~ ordinal level ω^(ω^ω^ω*2)
F[b]a[#+1+[#+#]]a ~ ordinal level ω^ω^(ω^ω + 1)
F[b]a[#+2+[#+#]]a ~ ordinal level ω^ω^(ω^ω + 2)
F[b]a[#+[#]+[#+#]]a ~ ordinal level ω^ω^(ω^ω + ω)
F[b]a[#+1+[#]+[#+#]]a ~ ordinal level ω^ω^(ω^ω + ω + 1)
F[b]a[#+[1+#]+[#+#]]a ~ ordinal level ω^ω^(ω^ω + ω2)
F[b]a[#+[2+#]+[#+#]]a ~ ordinal level ω^ω^(ω^ω + ω3)
F[b]a[#+[#+1]+[#+#]]a ~ ordinal level ω^ω^(ω^ω + ω^2)
F[b]a[#+[#+2]+[#+#]]a ~ ordinal level ω^ω^(ω^ω + ω^3)
F[b]a[#+[#+3]+[#+#]]a ~ ordinal level ω^ω^(ω^ω + ω^4)
F[b]a[#+1+#+#]]a ~ ordinal level ω^ω^(ω^ω*2)
F[b]a[#+2+#+#]]a ~ ordinal level ω^ω^(ω^ω*3)
F[b]a[#+3+#+#]]a ~ ordinal level ω^ω^(ω^ω*4)
F[b]a[#+#+1+#]]a ~ ordinal level ω^ω^ω^(ω + 1)
F[b]a[#+1+#+1+#]]a ~ ordinal level ω^ω^(ω^(ω + 1)*2)
F[b]a[#+#+2+#]]a ~ ordinal level ω^ω^ω^(ω + 2)
F[b]a[#+#+3+#]]a ~ ordinal level ω^ω^ω^(ω + 3)
F[b]a[#+#+[#]+#]]a ~ ordinal level ω^ω^ω^ω2
F[b]a[#+#+1+[#]+#]]a ~ ordinal level ω^ω^ω^(ω2 + 1)
F[b]a[#+#+2+[#]+#]]a ~ ordinal level ω^ω^ω^(ω2 + 2)
F[b]a[#+#+[1+#]+#]]a ~ ordinal level ω^ω^ω^ω3
F[b]a[#+#+[2+#]+#]]a ~ ordinal level ω^ω^ω^ω4
F[b]a[#+#+#+1]a ~ ordinal level ω^ω^ω^ω^2
F[b]a[#+#+1+#+1]a ~ ordinal level ω^ω^ω^(ω^2 + 1)
F[b]a[#+#+[#]+#+1]a ~ ordinal level ω^ω^ω^(ω^2 + ω)
F[b]a[#+#+[1+#]+#+1]a ~ ordinal level ω^ω^ω^(ω^2 + ω2)
F[b]a[#+#+[#+1]+#+1]a ~ ordinal level ω^ω^ω^(ω^2*2)
F[b]a[#+#+[1+#+1]+#+1]a ~ ordinal level ω^ω^ω^(ω^2*3)
F[b]a[#+#+#+2]a ~ ordinal level ω^ω^ω^ω^3
F[b]a[#+#+#+3]a ~ ordinal level ω^ω^ω^ω^4
F[b]a[#+#+#+#]a ~ ordinal level ω^ω^ω^ω^ω
F[b]a[#+#+#+1+#]a ~ ordinal level ω^ω^ω^ω^(ω + 1)
F[b]a[#+#+#+2+#]a ~ ordinal level ω^ω^ω^ω^(ω + 2)
F[b]a[#+#+#+[#]+#]a ~ ordinal level ω^ω^ω^ω^ω2
F[b]a[#+#+#+1+[#]+#]a ~ ordinal level ω^ω^ω^ω^(ω2 + 1)
F[b]a[#+#+#+[1+#]+#]a ~ ordinal level ω^ω^ω^ω^ω3
F[b]a[#+#+#+[2+#]+#]a ~ ordinal level ω^ω^ω^ω^ω4
F[b]a[#+#+#+#+#+1]a ~ ordinal level ω^ω^ω^ω^ω^2
F[b]a[#+#+#+#+#+2]a ~ ordinal level ω^ω^ω^ω^ω^3
F[b]a[#+#+#+#+#]a ~ ordinal level ω^ω^ω^ω^ω^ω
F[b]a[#+#+#+#+#+1]a ~ ordinal level ω^ω^ω^ω^ω^ω^2
F[b]a[#+#+#+#+#+#]a ~ ordinal level ω^ω^ω^ω^ω^ω^ω
F[b]a[#+#+#+#+#+#+#]a ~ ordinal level ω^ω^ω^ω^ω^ω^ω^ω
F[b]a[#+#+#+#+#+#+#+#]a ~ ordinal level ω^ω^ω^ω^ω^ω^ω^ω^ω
...
And the limit is: F[b]a[##]n = F[b]a[#+#+#+...+#+#+#]a w/ n #'s, which has the ordinal level ε0 in the fast-growing hierarchy, with respect to the Wainer hierarchy.
What if we continue? (Note: ordinal levels are given with respect to the existing, famous hierarchies (i.e. Veblen hierarchy, (extended) Buchholz hierarchy)):
F[b]a[##]a#a ~ ordinal level ε0+1
F[b]a[##]a#a#a ~ ordinal level ε0+2
F[b]a[##]a[1+#]a ~ ordinal level ε0+ω
F[b]a[##]a[1+#]a[1+#]a ~ ordinal level ε0+ω2
F[b]a[##]a[2+#]a ~ ordinal level ε0+ω^2
F[b]a[##]a[3+#]a ~ ordinal level ε0+ω^3
F[b]a[##]a[#+1]a ~ ordinal level ε0+ω^ω
F[b]a[##]a[1+[#+1]]a ~ ordinal level ε0+ω^(ω + 1)
F[b]a[##]a[1+#+1]a ~ ordinal level ε0+ω^ω2
F[b]a[##]a[#+2]a ~ ordinal level ε0+ω^ω^2
F[b]a[##]a[#+3]a ~ ordinal level ε0+ω^ω^3
F[b]a[##]a[#+#]a ~ ordinal level ε0+ω^ω^ω
F[b]a[##]a[#+#+1]a ~ ordinal level ε0+ω^ω^ω^2
F[b]a[##]a[#+#+#]a ~ ordinal level ε0+ω^ω^ω^ω
F[b]a[##]a[#+#+#+#]a ~ ordinal level ε0+ω^ω^ω^ω^ω
F[b]a[##]a[##]a ~ ordinal level ε0*2
F[b]a[##]a[##]a[##]a ~ ordinal level ε0*3
F[b]a[##]a[##]a[##]a[##]a ~ ordinal level ε0*4
F[b]a[1+[##]]a ~ ordinal level ε0*ω
F[b]a[2+[##]]a ~ ordinal level ε0*ω^2
F[b]a[3+[##]]a ~ ordinal level ε0*ω^3
F[b]a[#+[##]]a ~ ordinal level ε0*ω^ω
F[b]a[#+1+[##]]a ~ ordinal level ε0*ω^ω^2
F[b]a[#+#+[##]]a ~ ordinal level ε0*ω^ω^ω
F[b]a[#+#+#+[##]]a ~ ordinal level ε0*ω^ω^ω^ω
F[b]a[#+#+#+#+[##]]a ~ ordinal level ε0*ω^ω^ω^ω^ω
F[b]a[1+##]a ~ ordinal level ε0^2
F[b]a[1+[1+##]]a ~ ordinal level ε0^2*ω
F[b]a[#+[1+##]]a ~ ordinal level ε0^2*ω^ω
F[b]a[#+#+[1+##]]a ~ ordinal level ε0^2*ω^ω^ω
F[b]a[2+##]]a ~ ordinal level ε0^3
F[b]a[3+##]]a ~ ordinal level ε0^4
F[b]a[#+##]]a ~ ordinal level ε0^ω
F[b]a[#+1+##]]a ~ ordinal level ε0^ω^2
F[b]a[#+#+##]]a ~ ordinal level ε0^ω^ω
F[b]a[#+#+#+##]]a ~ ordinal level ε0^ω^ω^ω
F[b]a[##+1]a ~ ordinal level ε0^ε0
F[b]a[1+[##+1]]a ~ ordinal level ε0^ε0*ω
F[b]a[#+[##+1]]a ~ ordinal level ε0^ε0*ω^ω
F[b]a[##+[##+1]]a ~ ordinal level ε0^(ε0 + 1)
F[b]a[1+##+[##+1]]a ~ ordinal level ε0^(ε0 + 2)
F[b]a[2+##+[##+1]]a ~ ordinal level ε0^(ε0 + 3)
F[b]a[#+##+[##+1]]a ~ ordinal level ε0^(ε0 + ω)
F[b]a[#+#+##+[##+1]]a ~ ordinal level ε0^(ε0 + ω^ω)
F[b]a[1+##+1]a ~ ordinal level ε0^(ε0*2)
F[b]a[2+##+1]a ~ ordinal level ε0^(ε0*3)
F[b]a[#+##+1]a ~ ordinal level ε0^(ε0*ω)
F[b]a[#+#+##+1]a ~ ordinal level ε0^(ε0*ω^ω)
F[b]a[##+2]a ~ ordinal level ε0^ε0^2
F[b]a[1+##+2]a ~ ordinal level ε0^(ε0^2*2)
F[b]a[#+##+2]a ~ ordinal level ε0^(ε0^2*ω)
F[b]a[##+3]a ~ ordinal level ε0^ε0^3
F[b]a[##+4]a ~ ordinal level ε0^ε0^4
F[b]a[##+#]a ~ ordinal level ε0^ε0^ω
F[b]a[##+#+1]a ~ ordinal level ε0^ε0^ω^2
F[b]a[##+#+#]a ~ ordinal level ε0^ε0^ω^ω
F[b]a[##+#+#+#]a ~ ordinal level ε0^ε0^ω^ω^ω
F[b]a[##+##]a ~ ordinal level ε0^ε0^ε0
F[b]a[##+##+##]a ~ ordinal level ε0^ε0^ε0^ε0
F[b]a[##+##+##+##]a ~ ordinal level ε0^ε0^ε0^ε0^ε0
F[b]a[##+##+##+##+##]a ~ ordinal level ε0^ε0^ε0^ε0^ε0^ε0
...
F[b]a[[1+#]#]a ~ ordinal level ε1
F[b]a[[2+#]#]a ~ ordinal level ε2
...
F[b]a[[#+1]#]a ~ ordinal level εω
F[b]a[[#+2]#]a ~ ordinal level ε(ω^2)
F[b]a[[#+#]#]a ~ ordinal level ε(ω^ω)
F[b]a[[#+#+#]#]a ~ ordinal level ε(ω^ω^ω)
F[b]a[[##]#]a ~ ordinal level ε(ε0)
F[b]a[[[1+#]#]#]a ~ ordinal level ε(ε1)
F[b]a[[[#+1]#]#]a ~ ordinal level ε(εω)
F[b]a[[[#+#]#]#]a ~ ordinal level ε(ε(ω^ω))
F[b]a[[[##]#]#]a ~ ordinal level ε(ε(ε0))
F[b]a[[[[##]#]#]#]a ~ ordinal level ε(ε(ε(ε0)))
F[b]a[[[[[##]#]#]#]#]a ~ ordinal level ε(ε(ε(ε(ε0))))
...
F[b]a[#[#+1]]a ~ ordinal level ζ0
F[b]a[[1+#][#+1]]a ~ ordinal level ε(ζ0 + 1)
F[b]a[[#+1][#+1]]a ~ ordinal level ε(ζ0 + ω)
F[b]a[[#+#][#+1]]a ~ ordinal level ε(ζ0 + ω^ω)
F[b]a[[##][#+1]]a ~ ordinal level ε(ζ0 + ε0)
F[b]a[#[1+[#+1]]]a ~ ordinal level ε(ζ0*2)
F[b]a[#[2+[#+1]]]a ~ ordinal level ε(ζ0*3)
F[b]a[#[#+[#+1]]a ~ ordinal level ε(ζ0*ω)
F[b]a[#[##+[#+1]]a ~ ordinal level ε(ζ0*ε0)
F[b]a[#[1+#+1]]a ~ ordinal level ε(ζ0^2)
F[b]a[#[2+#+1]]a ~ ordinal level ε(ζ0^3)
F[b]a[#[#+2]]a ~ ordinal level ε(ζ0^ω)