User blog comment:Emlightened/Hypernomials (Γ₀)/@comment-5529393-20160113233321/@comment-27513631-20160114072601

I'd argue that $$\Gamma_0$$ can pretty much be reached by using the Veblen hierarchy directly, (and not using $$\lbrace0\rbrace$$, or 2 as the right argument), but the upper bound would be considerably harder. I mainly put it there because it's a decent lower bound, and the construction in question shouldn't have any function that dominates all of these.

When I originally considered the problem, I thought about the base being arbitrary, but it's both harder to reduce, and to order.