User blog comment:Revilo30703/Knuth's Up Arrow Infinite Expantion/@comment-27173506-20160123125539

This looks interesting, and a bit similar to this.

You didn't specify what happens when the array of the arrows isn't in the form of (2,A,c) where A is a series of ones. For dxample, how would you solve 3^(3,2)4? You say something about height and length, but I don't see any rules.

Infinities are indeed useful in googology. Look at the Fast-growing hierarchy. I believe that your notation, if properly defined, would reach \(\f_\omega^\omega(n)\).