I decided to extend my notation so it would be stronger.
Up to (^^(^^()^^)^^) operator[]
Operators up to (^^()^^) are defined as it follows.
Then:
a^(^^(n)^^)^b = a^(^(^(^(...(^(^(^(n)^)^)^)...)^)^)^)^b with n levels
a^(^^(^(^^(n)^^)^)^^)^b = a^(^^(^(^(^(...(^(^(^(n)^)^)^)...)^)^)^)^^)^b with n levels
and the new operator...
a^(^^(^(^(^^(n)^^)^)^)^b = a^(^^(^(^^(^(^^(...(^(^^(^(^^(^(^^(n)^^)^)^^)^)^^)^)...)^^)^)^^)^)^^)^)^^)^b with n levels
also:
a^(^^(^(^(^(^^(n)^^)^)^)^)^^)^b = a^(^^(^(^(...(^^(^(^(n)^)^)^^)...)^)^)^^)^b with n levels
etc.
Up to (^^^()^^^) operator[]
Then:
a^(^^(^^(n)^^)^^)^b = a^(^^(^(^(^(...(^(^(^(^^(n)^^)^)^)^)...)^)^)^)^^)^b with parentheses exponentiated n times
a^(^^(^^(^(^^(^^(n)^^)^^)^)^^)^^)^b = a^(^^(^^(^(^(^(...(^(^(^(^^(n)^^)^)^)^)...)^)^)^)^^)^b with parentheses exponentiated n times
a^(^^(^^(^(^(^^(^^(n)^^)^^)^)^)^^)^^)^b = a^(^^(^^(^(^^(^^(...(^(^^(^^(n)^^)^^)^)...)^^)^^)^)^^)^^)^b with n levels
etc.
With this pattern setted on, this continuation should be obvious:
a^(^^(^^(^^(n)^^)^^)^^)^b = a^(^^(^^(^(^(^(...(^(^(^(^^(n)^^)^)^)^)...)^)^)^)^^)^^)^b with n levels
a^(^^(^^(^^(^^(n)^^)^^)^^)^^)^b = a^(^^(^^(^^(^(^(^(...(^(^(^(^^(^^(^^(n)^^)^^)^^)^)^)^)...)^)^)^)^^)^^)^^)^b with n levels
etc.
Then:
Up to (^()^()^()^) operator[]
This part is a little longer.
We define:
a^(^^^(n)^^^)^b = a^(^^(^^(^^(...(^^(^^(n)^^)^^)...)^^)^^)^b with n levels
a^(^^^(^(^^(^^(n)^^)^^)^)^^^)^b = a^(^^^(^(^(^(...(^(^(^^(n)^^)^)^)...)^)^)^)^^^)^b with n levels
a^(^^^(^(^^(^^^(n)^^^)^^)^)^^^)^b = a^(^^^(^(^^(^^(^^(...(^^(n)^^)...)^^)^^)^^)^)^^^)^b with n levels
a^(^^^(^(^^(^^(^^^(n)^^^)^^)^^)^)^^^)^b = a^(^^^(^(^^(^^^(^^(^^^(...(n)...)^^^)^^)^^^)^^)^)^^^)^b with n levels
a^(^^^(^(^^^(n)^^^)^)^^^)^b = a^(^^^(^(^^(^^(^^(...(^^(^^(^^^(n)^^^)^^)^^)...)^^)^^)^^)^)^^^)^b with n levels
a^(^^^(^(^^^(^^(n)^^)^^^)^)^^^)^b = a^(^^^(^(...(^(^^^(n)^^^)^)...)^)^^^)^b with n levels
a^(^^^(^(^^^(^^(^^^(n)^^^)^^)^^^)^)^^^)^b = a^(^^^(^(^^^(^^(^^(^^(...(^^(^^(n)^^)^^)...)^^)^^)^^)^^^)^)^^^)^b with n levels
a^(^^^(^^(n)^^)^^^)^b = a^(^^^(^(^^^(^^(^^^(^^(^^^(...(n)...)^^^)^^)^^^)^^)^^^)^)^^^)^b with n levels
-- UNDER CONSTRUCTION --
FGH[]
a^(^^(n)^^)^b = w^2
a^(^^(^(^^(n)^^)^)^^)^b = w^2*2
a^(^^(^(^(^^(n)^^)^)^^)^b = w^3
a^(^^(^^(n)^^)^^)^b = w^w
NOTE: This series of arrows is getting random and it's possible that it will eventually go out of control. Please say me if you want me to close this project.
-- UNDER CONSTRUCTION --