Googolological notation 2

An improved version of Googolological notation

Two entry arrays
First if all {a,b} is equal to a^^^b.

Multi-entry arrays
Multi-entry arrays are solved recursively: {...,a,b} = {{...,a},b}.

Square brackets
[a,b] is equal to {a,a,...,a,a} with b a's

Square brackets can also be multi-entry arrays. Solve them like normal arrays

Higher levels
[...] becomes {...}2.

Then we define {...}3, it is to [...] as [...] is to {...}.

The f function
f(n) = {n,n}n.