This is the next, and strongest extension yet. It introduces {n} separators. {0} is the tilde, {1} is repeated {0} which is a pipe, {2} is repeated {1}, all the way to {n}.
The rules will be quite similar.
A,B... represents any separator, a separator being a string of tildes. Any {0}s should be replaced by tildes.
a~b = a^b
a~b~c = a^^^...c...^^^b = a{c}b
a A ... A b A 1 = a A ... A b
a A ... A b A 1 A c = a A ... Ab
a A ... A b B~ c = a A ... A b B b B~ (c-1) where B is any separator and B~ is B with a tilde at the end
a A ... A b B<n>c = a A ... A b B[c<n-1>] b where B is any separator, B<n> is B with an n-separator at the end, and [m<n>] is m copies of {n}
a~...~b~c~d = a~...~b~(a~...~b~(c-1)~d)~(d-1)
n{2}n ~ w^w^2
n{2}~n ~ w^(w^2+1)
n{2}~~n ~ w^(w^2+2)
n{2}|n ~ w^(w^2+w)
n{2}|~n = ~ w^(w^2+w+1)
n{2}||n ~ w^(w^2+w2)
n{2}{2}n ~ w^(2w^2)
n{2}{2}{1}n ~ w^(2w^2+w)
n{2}{2}{2}n ~ w^(3w^2)
n{3}n ~ w^w^3
n{4}n ~ w^w^4
n{n}n ~ w^w^w
The epsilon barrier is difficult to pass. I may never pass it. But I can rest knowing I at least reached low tetrational omegas, which beats many notations, and reaches all the way up to the (0,1) separator in BEAF.
Duhypergol = 10{2}100
Great Duhypergol = 10{2}100~2
Duhyperdol (Tilde-ex-hypergol) = 10{2}10~100
Pipe-ex-hypergol = 10{2}10|100
Pipe-tilde-ex-hypergol = 10{2}10|~100
Dupipe-ex-hypergol = 10{2}10||100
Trihypergol = 10{3}100
Terhypergol = 10{4}100
Pephypergol = 10{5}100
Dekhypergol = 10{10}100
Hecthypergol = 10{100}100 ~ Gongulus