Not what I meant, I meant with the +, ^, and stuff.
This hasn't been answered.
Ah yes, 4 more additions to the mix.
ε_n = (ε_n-1)^(ε_n-1)^(ε_n-1)^..., where ε_0 = ω^ω^ω^...(ω times)
n * 1 = n, therefore ω1 = ω * 1 = ω
No, this Omega 1.
No, the SPECIFIC ε_n.
Hello?
The specific value of n.
Please continue this.
ε1 = ε0^ε0^ε0^...
ε2 = ε1^ε1^ε1^...
ε3 = ε2^ε2^ε2^...
whatever you set the value of n to!
No,
the specific value of n used in
that's used for ζ_0.
ε_ε_0 = (ε_(ε_0-1))^(ε_(ε_0-1))^(ε_(ε_0-1))^...
ε_ε_ε_0 = (ε_ε_(ε_0-1))^(ε_ε_(ε_0-1))^(ε_ε_(ε_0-1))^...
do this process until you reach ζ_0
Oh. Ok.
PLEASE CONTINUE THIS AND FILL IN BLANKS! You can use
".
.
."
if you want.
_ represents underscore. Anyways, here is the extended version:
ω = ℕ
ω+1 = {1,2,3,4,...,inf-2,inf-1,1}
ω+2 = {1,2,3,4,...,inf-2,inf-1,1,2}
ω+n = {1,2,3,4,...,inf-2,inf-1,1,2,...,n-2,n-1,n}
ω2 = ω+ω = {1,2,3,4,...,inf-2,inf-1,1,2,...,inf-2,inf-1}
ω3 = ω+ω+ω = ω2 + ω = {1,2,3,4,...,inf-2,inf-1,1,2,...,inf-2,inf-1,1,2,...,inf-2,inf-1}
ω^2 = ωω = ω+ω+... (ω)
ω^3 = ωωω = ωω * ω = ωω ω = ω+ω+...(ω^2)
ω^ω = ωω ω*...(ω)
ω^ω^ω = ωω ω*...(ω^ω)
ε0 = ω^ω^ω^...(ω)
ε_1 = ε0^ε0^ε0^...
ε_n = (ε_n-1)^(ε_n-1)^(ε_n-1)^...
ζ_0 = ε_ε_ε_...(ε0)
φ(4,0) = ζ_ζ_ζ_...(ζ0)
φ(5,0) = φ(4,φ(4,...))
φ(n,0) = φ(n-1,φ(n-1,...))
φ(1,0,0) = φ(φ(...,0),0)
φ(2,0,0) = φ(1,φ(1,...,0),0)
φ(3,0,0) = φ(2,φ(2,...,0),0)
φ(1,0,0,0) = φ(φ(...,0,0),0,0)
φ(1,0,0,0,0) = φ(φ(...,0,0,0),0,0,0)
φ(1,0,0,0,0,0) = φ(φ(...,0,0,0,0),0,0,0,0)
θ(Ω^ω) = φ(1,0,0,0,...) (Small Veblen Ordinal)
This is where we start incorporating uncountable ordinals into countable ordinals. θ(β) where β is an uncountable ordinal or cardinal , it basically makes the uncountable ordinal countable. This is where things are getting more complex. We might not be able to continue further due to the complexity of the next ordinals (sorry if i sound like AI 
) However, there is way more after this point and the first one is called the Large Veblen Ordinal.
We should stop talking about transfinite ordinals and focus on the finite numbers called googology.
Ok. What's the closest number to Omega?
ω-1
All jokes aside, since EVERY number is closer to 0 than infinity (even the ill-defined ones) we cannot say things like the greatest finite number closer to infinity than zero although infinity-1 is technically a finite number.
I meant the closest KNOWN finite number other than the obvious ω-whatever stuff.
Large Number Garden Number. It is not computable with any ordinal in the Fast Growing Hierarchy. Not even the Bashicu Matrix System. I can't even screenshot the full definition.
Just do multiple screenshots.
Is it bigger than the Bachmann-Howard Ordinal?
WHAT! THAT DEFINITION IS HUGE! Also, since LNGN isn't bigger than the Bachmann-Howard Ondinal, that takes LNGN's place as the closest known finite number other than (Omega minus whatever).