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 --