This extension will introduce pipes to the mix.
a|b = a~~~...b...~~~a
a|~b = a|a|...b...|a|a
a||b = a|~~...b...~~a
a|||b = a||~~...b...~~a
Let's go!
A,B... represents any separator, a separator being a string of 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 B with a tilde at the end
a A ... A b B|c = a A ... A b B[c~] b where B| is B with a pipe at the end, and [n~] is n tildes
a~...~b~c~d = a~...~b~(a~...~b~(c-1)~d)~(d-1)
n|n ~ w^w
n|n~2 ~ w^w+1
n|n~n ~ w^w+w
n|n~n~n ~ w^w+w2
n|n~~n ~ w^w+w^2
n|n~~n~~n ~ w^w+2w^2
n|n~~~n ~ w^w+w^3
n|n|n ~ 2w^w
n|n|n|n ~ 3w^w
n|~n ~ w^(w+1)
n|~n|~n ~ 2w^(w+1)
n|~~n ~ w^(w+2)
n||n ~ w^w2
n|||n ~ w^w3
n||...n pipes...||n ~ w^w^2
Pipoogol = 10|100
Dipoogol = 10|10~100
Tripoogol = 10|10~10~100
Teripoogol = 10|10~10~10~100
Deutero-pipoogol = 10|10|100
Trito-pipoogol = 10|10|10|100
Tilded-pipoogol = 10|~100
Dutilded-pipoogol = 10|~~100
Piped-pipoogol = 10||100
Tilded-piped-pipoogol = 10||~100
Dupiped-pipoogol = 10|||100
Tripiped-pipoogol = 10||||100
Enenintaennapiped-pipoogol = 10||...[100 pipes]...||100