See:
This part is a part of a series of ideas I had. Reaching significantly past ψ(Ωω) is very hard.
EDIT: With some help with Googology Discord, I've improved my approximations.
EDIT 2: I just realized ψ(εΩ+1) is ψ(Ω2) in psi. This is more powerful than I thought after all. Ignore the title.
EDIT 3: I need to stop jumping to conclusions about approximating this.
See the rules from Part 2:
A ↑a+1 b = A ↑a A ↑a A ↑a ... ↑a A ↑a A ↑a A with b "A"s
A ↑n ★A = (A ↑n ★) ↑n A
UPGRADE:
<A,A> ↑a+1 b = <A,A> ↑a <A,A> ↑a <A,A> ↑a ... ↑a <A,A> ↑a <A,A> ↑a <A,A> with b "<A,A>"s
<A,A> ↑n ★A = (<A,A> ↑n ★) ↑n A
Recursive progression[]
<A,A>A ≈ ψ(Ωω)
<A,A>AA ≈ ψ(Ωω+1)
<A,A>AAA ≈ ψ(Ωω2)
<A,A>AA↑↑A ≈ ψ(Ωψ(Ω))
<A,A>AA ↑A A ≈ ψ(Ωψ(Ωω))
<A,A>A<A,A>A ≈ ψ(Ωψ(Ωω))
<A,A>AA ≈ ψ(ΩΩ)
<A,A>AAA ≈ ψ(ΩΩ+1)
<A,A>AAAA ≈ ψ(ΩΩ+ω)
<A,A>AA<A,A>A ≈ ψ(ΩΩ+ψ(Ωω))
<A,A>AA<A,A>AA ≈ ψ(ΩΩ2)
<A,A>AA ≈ ψ(ΩΩωω)
<A,A>A↑↑A ≈ ψ(ΩΩψ(Ω))
<A,A><A,A>A ≈ ψ(ΩΩ2)
<A,A><A,A>AA ≈ ψ(ΩΩω)
<A,A><A,A><A,A>A ≈ ψ(ΩΩΩ)
<A,A>↑↑A ≈ ψ(ΩΩ2)
(<A,A>↑↑A)A ≈ ψ(ΩΩ2+Ω)
(<A,A>↑↑A)<A,A>↑↑A ≈ ψ(ΩΩ22)
<A,A>↑↑AA ≈ ψ(ΩΩ22)
<A,A>↑↑AA ≈ ψ(ΩΩ2ω)
<A,A>↑↑<A,A>A ≈ ψ(ΩΩ2Ω)
<A,A>↑↑<A,A>↑↑A ≈ ψ(ΩΩ2Ω2)
<A,A>↑↑↑A ≈ ψ(ΩΩ3)
(<A,A>↑↑↑A)↑↑A ≈ ψ(ΩΩ3Ω2)
<A,A>↑↑↑AA ≈ ψ(ΩΩ32)
<A,A>↑↑↑<A,A>↑↑A ≈ ψ(ΩΩ3Ω2)
<A,A>↑↑↑<A,A>↑↑↑A ≈ ψ(ΩΩ3Ω3)
<A,A>↑↑↑↑A ≈ ψ(ΩΩ4)
<A,A>↑↑↑↑↑A ≈ ψ(ΩΩ5)
<A,A> ↑A A ≈ ψ(ΩΩω)
So after that, <<A,A>,A> is easily definable similarly to <A,A> and can be iterated similarly.
<<A,A>,A>A ≈ ψ(ΩΩω)
<<A,A>,A><<A,A>,A>A ≈ ψ(ΩΩψ(ΩΩω))
<<A,A>,A>A ≈ ψ(ΩΩΩ)
<<A,A>,A>↑↑A ≈ ψ(ΩΩΩ2)
<<A,A>,A>↑↑↑A ≈ ψ(ΩΩΩ3)
<<A,A>,A> ↑A A ≈ ψ(ΩΩΩω)
<<<A,A>,A>,A>A ≈ ψ(ΩΩΩΩ)
Limit: <<<...<<<A,A>,A>,A>...A>,A>,A> ≈ ψ(omega fixed point ) (commonly denoted as ψ(ψI(0)) but pbot doesnt like that smh)
This isn't over.