User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-1605058-20130718061544/@comment-30118230-20170202183708

It can go all the way if you do it infinitely,but for any finite amount of recursive diagonalizing,you can never reach the Church-Kleene ordinal.We can also diagonalize over those diagonalizers and that notation already exists.