Omegaggol!

Define the fundamental sequences!

Γ₀ is the [[Feferman–Schütte ordinal]], i.e. it is the smallest α such that φ_α(0) = α.
For Γ₀, a fundamental sequence could be chosen to be \(\Gamma_0 [0] = 0\) and \(\Gamma_0 [n+1] = \varphi_{\Gamma_0 [n]} (0) \,.\)
For Γ_β+1, let \(\Gamma_{\beta+1} [0] = \Gamma_{\beta} + 1 \) and \(\Gamma_{\beta+1} [n+1] = \varphi_{\Gamma_{\beta+1} [n]} (0) \,.\)
For Γ_β where \(\beta < \Gamma_{\beta} \) is a limit, let \(\Gamma_{\beta} [n] = \Gamma_{\beta [n]} \,.\)

We can now create a FIH with the base rule \(g_0(n) = \Gamma_{n^n}\) (I use g to prevent ambiguity.)
Then, the Large Gamma Ordinal is defined as \(g_{10^100}(10^{100})\) We hereafter call this the LGO.
Finally, Omegaggol is \(f_{LGO}(10^{100})\) in the FGH.