User blog comment:MachineGunSuper/The Hierarchy Replacement Function/@comment-30754445-20171222140917

It's a very good idea.

In fact, it's such a good idea, that some of the people here tried to implement it. But it is very very very difficult to pull through (or in your own words: to "use it correctly").

It's also similar to what what Johnathan Bowers (one of the fathers of googology) tried to do with is array notation. He used the symbol "&" to denote the kind of thing you're talking about here, so in his notation your example would look like this:

666↑↑666 & 666

Which is indeed close to fε₀(666)

Unfortunately, his & operator breaks down when we have more than two arrows.

At any rate, if done correctly, this will have the strength of ψ(Ωω) (that's the first ordinal where the SGH and FGH have the same growth rate). You can probably guess that this is tremendously strong even by googological standard (it's somewhere around class 160 on my Psi Levels Scale).