User blog comment:Simplicityaboveall/The Construction of Extremely Large Numbers/@comment-24920136-20160725013617/@comment-1605058-20160725180402

@Chronolegends Assuming OP is using the same notation as e.g. this page (which agrees with the use of term "hereditary form"), then $$R_{10}^\omega(10^{10})=\omega^\omega$$, and indeed $$R_{10}^\omega$$ is a function values of which have supremum $$\varepsilon_0$$.

If, however, he isn't using the same function (which is very likely the case, since he seems to imply $$R_{10}^\omega$$ takes values larger than $$\varepsilon_0$$), then I completely agree that there is a lot left to define.