User blog comment:Emlightened/Hypernomials (Γ₀)/@comment-5529393-20160113233321

This is a very interesting problem! Skolem created a similar family of functions, except instead of general Knuth arrows it was limited to exponentiation only. He then asked what was the order type of that family under eventual domination. This problem is still unsolved.

Hardy proved that the above family is linear under eventual domination, and Ehrenfeucht proved that it was a well-order using Kruskal's theorem. Levitz proved that the order type was at most $$\varphi(2,0)$$. A couple of other papers proved that $$ 2^x$$ and $$2^{2^{x}}$$ were at $$\omega^\omega$$ and $$\omega^{\omega^\omega}$$ respectively. That seems to be all that is known about the problem.

I imagine that your family could be proven to be a well-order in a manner similar to Ehrenfeucht's, and one could prove that the order is at least $$\Gamma_0$$, by constructing an injection from $$\Gamma_0$$ into the family, and proving that the injection preserves order. I imagine that proving that the order type is precisely $$\Gamma_0$$ is an enormously difficult problem, however.