User blog comment:Ubersketch/Lexicographic ordinal notation up to the Bachmann-Howard ordinal/@comment-30869823-20190913225449/@comment-39541634-20190916171457

Because there's no way to generate the ordinal "Ωω" (or anything larger) in this system. The fact that you can use ψ doesn't help, because ψ(anything) is always countable. ψ(Ω) - for example - is equal to ε₀. Since you have addition, you can also have things like ψ(Ω+Ω)=ε₁ and ψ(Ω+Ω+Ω)=ε₂ and so on... but after that, the notation gets stuck.

To go beyond this, there are a couple of possible approaches:

(1) Allow multiplication and exponention, as Fejfo has suggested.

-or-

(2) Add another function ψ₁ and another cardinal Ω₂. The ψ₁ function is constructed in such a way that it generates ordinals which are larger than Ω but smaller than Ω₂. Then we could easily generate Ωω by writing ψ₁(0). And ε_ω would then be written as ψ(ψ₁(0)).

The limit of the first appraoch is exactly the BHO. The second approach gets you a bit further (with the BHO being equal to either ψ(Ω₂) or ψ(ψ₁(Ω₂)), depending on the precise way the OCF was set up)