See here.
MOST, or Multiarrowed Ordinal Snowballing Train, is a simple ordinal notation created by the once sane Edwin Shade.
It is defined as follows:
α ↑γ β = α ↑γ-1 (α ↑γ (β - 1) + 1) if β is the successor of a limit ordinal (so as to not choke on ε0)
α ↑γ β = α ↑γ-1 (α ↑γ (β - 1)) if β is a successor ordinal but the previous rule does not apply
α ↑γ β = sup{α ↑γ γ|γ < β}
α ↑γ β = sup{α ↑δ β|δ < γ} if γ is a limit ordinal
Countable Index
ω↑↑ω = ε0
ω↑↑(ω+1) = ω↑(ε0+1)
ω↑↑(ω+n) = ω↑ω↑ω↑...↑ω↑ω↑ω↑(ε0+1) (n "ω"s)
ω↑↑(ω2) = ε1
ω↑↑(ωn) = εn-1
ω↑↑(ω2) = εω
ω↑↑(ω2+ω) = εω+1
ω↑↑(ω2+ωn) = εω+n
ω↑↑(ω22) = εω2
ω↑↑(ω2n) = εωn
ω↑↑(ω3) = εω2
ω↑↑(ωn) = εωn-1
ω↑↑(ωω) = εωω
ω↑↑↑3 = εε0
ω↑↑↑n = εε...ε0 (n-1"ε"s)
ω↑↑↑ω = ζ0
ω↑↑↑↑ω = η0
ω{n}ω = φ(n-1,0)
ω{ω}ω = φ(ω,0)
ω{ω{ω}ω}ω = φ(φ(ω,0),0)
Limit of countable-indexed MOST is Γ0. Edwin then defines ω{Ω}ω as the fixed point of α ↦ ω{α}ω.
First Uncountable Cardinality
ω{Ω}ω = Γ0 = φ(1,0,0)
ω{Ω}(ω2) = Γ1 = φ(1,0,1)
ω{Ω+1}ω = φ(1,1,0)
ω{Ω+2}ω = φ(1,2,0)
ω{Ω+α}ω = φ(1,α,0)
ω{Ω2}ω = φ(2,0,0)
ω{Ωα}ω = φ(α,0,0)
ω{Ω2}ω = φ(1,0,0,0)
ω{Ω3}ω = φ(1,0,0,0,0)
ω{Ωω}ω = θ(Ωω)
ω{ΩΩ}ω = θ(ΩΩ)
ω{Ω↑↑ω}ω = θ(εΩ+1)
ω{Ω↑↑↑ω}ω = θ(ζΩ+1)
ω{Ω{ω}ω}ω = θ(φ(ω,Ω+1))
Up until this point, it's clear that ω{α}ω = θ(α).
ω{Ω{Ω}ω}ω = θ(Ω2)
ω{Ω{Ω{Ω}ω}ω}ω = θ(Ω3)
Limit: θ(Ωω). At this point, we can define Ωα as the fixed point of β ↦ Ωα-1{β}ω.
Beyond omega one
It's at this point that I'm not entirely certain how to work with the ordinal notations, so bear with me--this could be entirely wrong.
ω{Ω2}ω = θ(Ωω)
ω{Ω2}α = θ(Ωα)
ω{Ω2+1}ω = θ(ΩΩ)
ω{Ω2+1}(ω+1) = θ(Ωθ(ΩΩ))
ω{Ω2+1}(ω2) = θ(ΩΩ2)
ω{Ω2+1}(ω2) = θ(ΩΩω)
ω{Ω2+1}(ω3) = θ(ΩΩω2)
ω{Ω2+1}(ωω) = θ(ΩΩωω)
ω{Ω2+2}ω = θ(ΩΩΩ)
ω{Ω2+3}ω = θ(ΩΩΩΩ)
ω{Ω2+ω}ω = ψ(ψ𝐈(0))
ω{Ω2+ω}(ω2) = ψ(ψ𝐈(1))
ω{Ω2+ω}(ω2) = ψ(ψ𝐈(ω))
ω{Ω2+ω}(ωω) = ψ(ψ𝐈(ωω))
ω{Ω2+ω}ω{Ω2+ω}ω = ψ(ψ𝐈(ψ(ψ𝐈(0))))
ω{Ω2+ω+1}ω = ψ(ψ𝐈(Ω))
ω{Ω2+ω+1}(ω2) = ψ(ψ𝐈(Ωω))
ω{Ω2+ω+2}ω = ψ(ψ𝐈(ΩΩ))
ω{Ω2+ω+3}ω = ψ(ψ𝐈(ΩΩΩ))
ω{Ω2+ω2}ω = ψ(ψ𝐈(ψ𝐈(0)))
ω{Ω2+ω3}ω = ψ(ψ𝐈(ψ𝐈(ψ𝐈(0))))
ω{Ω2+ω2}ω = ψ(𝐈)
ω{Ω2+ω2}(ω2) = ψ(𝐈×ω)
ω{Ω2+ω2+1}ω = ψ(𝐈×Ω)
ω{Ω2+ω2+2}ω = ψ(𝐈×ΩΩ)
ω{Ω2+ω2+ω}ω = ψ(𝐈×ψ𝐈(0))
ω{Ω2+ω2+ω2}ω = ψ(𝐈×ψ𝐈(ψ𝐈(0)))
ω{Ω2+ω22}ω = ψ(𝐈2)
ω{Ω2+ω3}ω = ψ(𝐈ω)
ω{Ω2+ωn}ω = ψ(𝐈ωn-2)
And from this point on ω{Ω2+α}ω = ψ(𝐈α) for α ≥ ωω. So the limit of MOST is:
ω{ψ𝐈(0)}ω = ψ(𝐈ψ𝐈(0))