User blog comment:Ikosarakt1/Ideas of going past psi(K)/@comment-68.111.90.23-20140625141707

I think what you may be missing is that the weakly compact notation actually already uses a ternary function.

Ψ(a, b, c) is the collapse of the ordinal a to an ordinal of type b, with inductive level c.

So Ψ(a, 0, c) is the normal collapsing function ψ(a, c).

Ψ(a, 1, c) is the collapsing function to the inaccessible hierarchy, so it is roughly parallel with χ.

Ψ(a, 2, c) is the collapsing function to the Mahlo hierarchy.

So I think Ψ(a, b, c) already encapsulates what you mean by χa(b, c, d).

The natural extension of the weakly compact notation would extend to indescribables, and would involve adding another variable Ψ(a, b, c, d). Because of this, I don't think it is all that clear what the extension of the sequence Ω, M, K, is.