User blog comment:TheKing44/The analytical beaver functions/@comment-39605890-20190605124029/@comment-35470197-20190605222759

I sometimes consider formulae in the language of first order set theory whose quantifiers are bounded by \(V_{\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}}\) so that it is definable in ZFC set theory. Although I have nver created an explicit number, it will be greater than the "\(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}\)-th order arithmetic" version of your definition if appropriately defined.

Also, recently I used formulae in the language of first order set theory satisfiable by a segment of \(V\) here. So my approach is not restricting the language itself, but restricting the domain.