User blog comment:Edwin Shade/A Small Question/@comment-30754445-20171029013412/@comment-30754445-20171029021538

Yup.

For example, look at the fastest growing function which can be programmed into a Turing Machine (a kind of abstract computer) with n states. The ordinal corresponding to the growth rate of this function can be set as the nth member of our fundamental sequence.

And the limit of this these ordinals would, indeed, be equal to ω₁ck.

The catch is that there's no way to actually compute this fundamental sequence, or any other fundamental sequence of ω₁ck. That's one of the things that "nonrecursive" means.