User blog comment:ANDY3556/Andrew's Improved A-Notation/@comment-31867576-20180611175616/@comment-5529393-20180611183657

A(x) = "x x-ated to itself x times" is already at the Ackermann level, or at the level of w in the fast-growing hierarchy. So much more than tetrational.

A(x,x) iterates A(x) x times, so it is about f_{w+1} in the FGH.

A(x,y,z), A(x,y,z,w), and so on don't do much, since you are just nesting A(x) y+z+w times. Similarly for things like A * (x,y,z).

A2(x) is a little less than A(A(...(A(x)...)) with A(x)+1 nestings, so it is about f_{w+1}(f_w(x)). Then A3(x) is about f_{w+1}(f_{w+1}(f_w(x))), and so on, so Ax(x) will have about x iterations of f_{w+1}(x), making it about f_{w+2}(x).