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

"plugging" a function that reaches f_δ into the f'_0 of a new hierarchy just allows you to reach f_δ+δ(n) where x is the previous ordinal you already reached. The limit of repeatedly looping in this way is f_δ*ω.for finitely many function loops f'_0, f_0, f'_0... etc.

You could invent some form of transfinite loop control but that would not be necesarily stronger or more elegant than what you are already doing.