User blog comment:Flavio61/Louis Epstein number list/@comment-27173506-20160118190252/@comment-27173506-20160119190223

I'm not "looking" for anything, I'm telling you the facts. Nesting on the bound is definitely stronger than nesting on the base, but nesting on the nestings is way, way stronger than either of them.

Furthermore, n xu n can be approximated (this is actually a lower bound, but they have approximately the same growth rate) as n^^^x+1. Here's why:

Firstly, n u n~n^^n=n^^^2. Furthermore, n uu n=n u n u n~n^^(n^^n)=n^^^3. In general, n xu n=n u n u n u n...n u n (with x u's)~n^^n^^n^^n...n^^n (with x+1 n's)=n^^^x+1.

So while 3 uuu 3 is greater than 3^^^3, 4 uuuu 4 is nothing compared to 4^^^^4.

Oh, by the way, here's a nice notation built on repeated exponentiation which is not exactly equal to the hyperoperators. It's slightly different than yours, but I thought you might be interested.