User blog comment:Ubersketch/A better Fast Growing Hierarchy: The Uber Hierarchy/@comment-30754445-20180922205035

The FGH is pretty much limitless, so any attempt to "extend" it would be - by definition - a naive extension.

It should be noted that the reason that the FGH is so strong, is that it doesn't really work on its own. It's just a framework. To get actual numbers from the FGH, we need to explicitly state the fundamental sequences we're using. The ordinals themselves are not enough.

At the low levels, this distinction between "ordinals" and "the fundamental sequences" might seem a little silly. Everybody knows that f   ω     (n)=f    n     (n) (after all, what else could it be?). But as the numbers go higher and the ordinals become more complex, this task of finding the fundamental sequences becomes harder and harder. Soon it becomes the crux of the challange (and at some point below PTO(Z2), the task becomes so difficult that no person alive knows how to go any further).

Adding seperators doesn't change this. Anything you can do with seperators, you can do with ordinals. In fact, every array notation that is worth its salt is, at its heart, an ordinal notation. From Saibian's E to Bird Arrays to Hollom's HAN to Hyp Cos' SAN - all these notations were consciously designed by their authors to behave like ordinals.