Interesting, so there are degrees such that some function of this degree isn't bounded by computable function, but there is no dominant function, and e.g. low c.e. sets have this property.

So let me get this straight - if there is $A$-computable function which isn't bounded by any computable function, then degree of $A$ is hyperimmune, and "most" of degrees are hyperimmune, in particular all recursively enumerable degrees are hyperimmune. Is this right?

@JoelDavidHamkins Do you think we could make a similar construction, so that we also have, for all ordinals $\alpha<\beta\leq\varepsilon_0$, that $F_\alpha$ is eventually outgrown by $F_\beta$? (that is, for sufficiently large $n$, $F_\alpha(n)<F_\beta(n)$? Your construction satisfies it for all ordinals other than $\varepsilon_0$ itself.

This is the best answer I could hope for! It's not only extremely simple, but modifies only very little number of fundamental sequences. Thanks!

Isn't this by definition $\Sigma_3$-admissible ordinal?

How is the description of these polygons given for the algorithm?

Thanks, Joel, this is exactly what I was looking for! I have yet to read the paper, but I'm sure many interesting results are given there.

@LiangYu I don't see why this should be a contradiction. It can be the case that the subset of $\Bbb N^U$ becomes finite.