User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-35870936-20180623152451

A lot of "large number" notations are basically ordinal notations, but with some extra element to convert them to a number. Take BMS as an example. If you have an expression like (0,0)(1,1)(2,2)(3,1), this represents an ordinal (in this case psi(e(W*2))). However, adding a pair of [] at the end will turn it into a large number, such as (0,0)(1,1)(2,2)(3,1)[5]. In this regard, BMS could be considered a very powerful ordinal notation. However, I'm pretty sure that TON is stronger than BMS.

In terms of OCF-type notations, "standard" notation gets up to psi(e(K+1)), and it is 100% well-defined. UNOCF gets higher than the limit of DAN, and I think UNOCF could be reasonably well-defined, but nobody has made a ruleset yet. However, the behaviour of UNOCF is quite easy to understand so I think it shouldn't be too hard to create a ruleset.

Another collapsing function is Aarex's OCF (which he is developing in Discord). It goes past UNOCF and DAN, but I have no idea of it's well-definedness.