User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-1605058-20180623095923

Taranovsky's C is conjectured (by Taranovsky himself) to be well-defined and to be strong enough to express proof-theoretic ordinals of ZFC and many of its extensions. If this were the case, then it would be the strongest ordinal notation by a long shot. However, the former has only been proven partially by Taranovsky (and I don't think a critical look has been taken at the proof) and I believe mathematicians don't really believe it is as strong as Taranovsky claims, but we have no proof whatsoever this way or another.

For notations actually recognized by other mathematicians, the strongest published one is probably the one in Rathjen's "An ordinal analysis of parameter free \(\Pi^1_2\)-comprehension", which describes the proof-theoretic ordinal of \(\Delta^1_2-CA+BI+\) parameter free \(\Pi^1_2\)-comprehension. However, back in 1995 Rathjen has announced a full ordinal analysis of \(\Pi^1_2-CA_0\) (and even \(\Delta^1_3-CA\)) but it has yet to appear. Some details are given in the paper "Recent Advances in Ordinal Analysis: \(\Pi^1_2-CA_0\) and related systems".