User blog comment:MachineGunSuper/The Hierarchy Replacement Function/@comment-32783837-20171223145318/@comment-30754445-20171223211355

Actually, ArtismScrub's definitions get stuck at  ε0:

ω↑↑(ω+1) = ω↑(ω↑↑ω) = ω↑ε0 = ε0

There are other ways to do this, though.

We could use down-arrows (iterated powers solved from left to right). These grow slower at first, but they escape the fixed-point problem because we get ε0↑ω instead of ε0↑ω. They're a bit of a nightmare to evaluate, though.

Or we could use Saibian's climbing method from his Hyper-E, that get us a whopping gamma-0 for ω↑↑↑ω (!).

But yes, when people say that this is very difficult, they aren't kidding.