User blog comment:Edwin Shade/How High Can You Go ?/@comment-5529393-20170920232027

Just saw this post - great to see someone new getting into googology! The obvious answer is that you can keep on going past $$\psi(\Omega^{\Omega^\Omega})$$ - next there's $$\psi(\Omega^{\Omega^\Omega})+\alpha$$, then eventually you reach $$\psi(\Omega^{\Omega^\Omega}+\alpha)$$, then $$\psi(\Omega^{\Omega^\Omega+\alpha})$$, then $$\psi(\Omega^{\Omega^{\Omega+\alpha}})$$, and eventually you get too $$\psi(\Omega^{\Omega^{\Omega^\Omega}})$$. The eventual limit is $$\psi(\Omega^{\Omega^{\Omega^{\Omega^\cdots}})$$, a famous ordinal called the Bachmann-Howard ordinal. If you understand how the $$\psi$$ function works, you can get at least this far; if not, I would be glad to answer any questions you might have. A good resource is this Wikipedia page; I've heard several people rave about this article. I see you've found the video series by Mark Giroux; another good Youtube series is this video series by David Metzler. I find Metzler to be a better explainer than Giroux, although his series is rather long. Finally, you can read my series of blog posts on ordinal notations, starting from this one.

Beyond the Bachmann-Howard ordinal, you can start collapsing larger cardinals, so you start seeing things like $$\Omega_\alpha$$ and $$\psi_\alpha$$. All four of the above resources talk about this' however, Giroux defines things incorrectly, so I don't recommend his videos on the subject. The Wikipedia page is rather short on this subject, so probably the Metzler videos and my blog posts are the way to go at this point.

To go even further beyond, you can start collapsing weakly inaccessibles, $$\alpha$$-weakly inaccessibles, hyper-inaccessibles. Then you can use weakly Mahlo cardinals to collapse to these various inaccessibles, then use weakly compact cardinals to collapse to various levels of Mahlo. My blog posts are the only one of the above resources that go beyond the $$\Omega$$ function, but you can also read the Wiki pages Ordinal notation and List of systems of fundamental sequences.

You can keep going beyond weakly compacts, but I will leave that for later. ;)