Hyper-Tilde notation is my notation. That's about it.
The first part of my notation is equivalent to chained-arrow notation
a~b = a^b
a~b~c = a^^^...c...^^^b = a{c}b
a~...~b~1 = a~...~b
a~...~b~1~c = a~...~b
a~...~b~c~d = a~...~b~(a~...~b~(c-1)~d)~(d-1)
The limit of this notation is omega-squared. Here's an analysis compared to BEAF and FGH.
Tilde Notation ~ BEAF ~ FGH ordinal
n~n ~ {n,n} ~ 2
n~n~2 ~ {n,n,2} ~ 3
n~n~n ~ {n,n,n} ~ w
n~n~n~2 ~ {n,n,1,2} ~ w+1
n~n~n~3 ~ {n,n,2,2} ~ w+2
n~n~n~n ~ {n,n,n,2} ~ w2
n~n~n~n~2 ~ {n,n,1,3} ~ w2+1
n~n~n~n~n ~ {n,n,n,3} ~ w3
n~n~n~n~n~n ~ {n,n,n,4} ~ w4
n~n~n~n...n~n~n with n copies of n ~ {n,n,n,n} ~ w^2
Tildoogol = 10~100
Great Tildoogol = 10~100~2
Great Great Tildoogol = 10~100~3
Dutildoogol = 10~10~100
Tritildoogol = 10~10~10~100
Tertildoogol = 10~10~10~10~100