User blog comment:Wythagoras/Extension of the busy beaver function/@comment-1605058-20131023174307/@comment-5529393-20131024024709

$$\omega_\Omega^{\text(CK)} = \Omega$$. Clearly it would be at least $$\Omega$$, and the fact that the admissible ordinals are unbounded in $$\Omega$$ shows that it is exactly $$\Omega$$.