User blog comment:Bubby3/Guesses about the strength of BMS and the Catching Hierarchy./@comment-32697988-20180419222547/@comment-32697988-20180420010938

I don't understand what you mean by stable ordinals, but do you mean limit of KP+Π_n-reflection? If so I do think (0,0,0)(1,1,1)(2,2,0) reaches that. Or even (0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,1) or something like that. Further analysis is nearly impossible due to the lack of an easy-to-understand ordinal notation, but I think the fanmade OCF project can give a solution.

You also talked about you expecting BMS not to be strong because of hydras, but DAN is a hydra as well. I don't think being a hydra will make it not so strong, and I personally think BMS is much stronger than what is shown here, although I have no reasons to justify this.