Ytukyg

I have not previously seen anything online that dives deeply into On:

In Conway's notation On denotes the ordinal numbers (and No denotes the set of all surreal Numbers). Basically the elements of On are just von Neumann ordinals. -Source

I would appreciate feedback on the following attempt to write large countable ordinals (& the functions that generate them) in Conway notation (my primary source of information in creating these constructions was Large Countable Ordinals):

epsilon-nought
$$varepsilon_{0}={omega,omega^omega,omega^{omega^omega},...|}$$
veblen function
$$phi={omega,varepsilon_0,zeta_{alpha},...|}$$
veblen hierarchy
$$phi_{gamma}(alpha)={phi_{0}(alpha)=omega^{alpha}, phi_{1}(alpha)=varepsilon_{alpha}, phi_{2}(alpha)=zeta_{alpha}...|}$$
feferman-schutte ordinal
$$Gamma_0=phi_{Gamma_0}(0)={phi_0(0),phi_{phi_0(0)}(0),phi_{phi_{phi_0(0)}(0)}(0),...|}$$
small veblen ordinal
$$SVO={phi_1(0), phi_{1,0}(0), phi_{1,0,0}(0),...|}$$
large veblen ordinal
$$LVO=psi(Omega^{Omega^Omega})={[???]...|}$$
bachmann-howard ordinal
$$BHO=psi(varepsilon_{Omega+1})={psi(Omega),psi(Omega^Omega),psi(Omega^{Omega^Omega})|}$$

Additionally, any online resources related to On would be greatly appreciated.

I have to say I know as little about large countable ordinals as I do about games. I actually didn't know the ordinal $Gamma_0$ was thought of as the first impredicative ordinal, had a name and so on.

Regarding your definitions, the function $phi_{gamma}(alpha)$ should also be greater than every ordinal $phi_{eta}^{circ n}(phi_{gamma}(beta)+1)$ for $eta < gamma$, $n in mathbb{N}$ and $beta<alpha$. I am not sure about what you mean by Vleben function, and I don't know about SVO, LVO, BHO.

Perhaps something you might find interesting is a phenomenon noticed by Conway and expended upon by Gonshor: the functions $phi_{gamma}$ can be extended to $mathbf{No}$ in a natural way.

For $x={L | R} in mathbf{No}$, you must know about $omega^x=phi_0(x)={0,mathbb{N} phi_0(L) | 2^{-mathbb{N}} phi_0(R)}$.
Then the class of numbers $e$ such that $omega^e=e$ is parametrized by $varepsilon_x=phi_1(x):={phi_0^{circ mathbb{N}}(0),phi_0^{circ mathbb{N}}(phi_1(L)+1) | phi_0^{circ mathbb{N}}(phi_1(R)-1)}$,
and one can keep going on. At every stage $0<gamma$, the function $phi_{gamma}$ parametrizes the class of numbers $e$ with $forall eta < gamma,phi_{eta}(e)=e$.

As for sources on $mathbf{On}$, since this is just the class of ordinals, you can just look into this. I don't know that new insight on ordinal numbers has been gained by seeing them as surreal numbers, at least not in a significant way.