User blog comment:QuasarBooster/Googology in Geometry/@comment-35470197-20191123001017

> It can describe the entire field of quadratically closed rational numbers

Do you mean that you are considering all constructible numbers? Then I guess that we also have a method to construct 1/(almost integer - integer), but a function derived from it is bounded by q^κ, where κ is the irrational degree of the almost integer. Therefore I do not idea on how to go beyond 2^{2^n}, either.