User blog comment:Denis Maksudov/Second M-notation/@comment-24920136-20160610211858/@comment-24920136-20160612165015

Using a new set of extensions such as superscripts, stacking array rules ontop of eachother.. etc.. looks good on paper, but you only get a growth of δ ---> δ2 for every recursion, so the limit to what you intend to do is f_δ*λ(n), where δ is whatever previous ordinal you reached with fδ(n)  and λ being whatever ordinal you would reach if you just started the extending process with the original fgh_0(n)

although my curiosity is piqued by how you entwined the nestings, and could actually reach some nice high places with more of a sketched concept.

I have toyed with the idea of having two functions reducing to eachother before but i haven't come up with anything solid as of yet.