User blog comment:GamesFan2000/If Ordinals used the entire hyper-operator system instead of just the mainstream hyper-operators/@comment-30754445-20190217074139

Unfortunately (and counter-intuitively), the usual set of hyper-operators won't give you bigger ordinals than exponentiation:

ω↑↑[anything >ω] =  ε₀

ω↑↑↑[anything >1] =  ε₀

ω↑↑↑↑[anything >1] = ε₀

This is because ε₀ has the strange property that ε₀ = ωε₀ (that's not a typo).

Due to this quirk, we have (for example)

ω↑↑(ω+1) = ωω↑↑ω = ωε₀ = ε₀ = ω↑↑ω

So we are stuck.

One way around this is to use down-arrows instead of up-arrows. Then we'd have:

ω↓↓↓ω = ε₀ (yes, that's three arrows rather than two)

ω↓↓↓(ω+1) = ε₀↓↓ω = ε₀ε₀ ω

ω↓↓↓(ωx2) = ε₁

and so on.

In this system you don't get "stuck" at ε₀. If I correctly remember an analysis I did a long time ago, the limit of ω↓....↓ω is φ(ω,0). Using your idea with down-arrows, we'll have (I think):

a₀ = ω

a₁ = φ(ω,0)

a₂ = φ(φ(ω,0),0) aa₀ = Γ₀

aa a₀ = ΓΓ₀

b₀ = φ(1,1,0)

And after that, it isn't clear what your definitions say we should do to get b₁, so I'll stop here.