Based on [this][1] answer. Working in ZF-Reg., is it possible to have a non-well founded set that is strictly supernumerous to any well founded set?

If that is possible, then can this be extended indefinitely? That is lets consider the well founded realm as the hierarchy whose base (first stage) is the empty set, and for the pusposes of the development I'm intending here lets phrase the base set of that hierarchy (which is the empty set) as the set of all sets that are hereditarily strictly subnumerous to empty set, now can we have a hierarchy whose base is the set of all herediarily strictly subnumerous to singleton sets, such that this base set is strictly supernumous to every set in the first hierarchy (whose base is the empty set), and that in turn also every set in that hierarchy itself is strictly subnumerous to the set of all hereditarily strictly subnumerous to doublton sets, and so on... That is, for every hierarchy whose base is the set of all hereditarily strictly subnumerous to $|x|$ sets, then all sets in that hierarchy are strictly subnumerous to the set of all hereditarily strictly subnumerous to $|y|$ sets, for any set $y$ that is strictly supernumous to $x$. 

Of course the idea is that each base set is non-well founded, it can contain sets having cyclic membership, and that what cause them to have their cardinality unleashed in such a manner.

Can we have such a model of ZF-Reg.??

Of course we need to add to ZF-Reg. the axiom that for every set $x$ there is a set of all sets hereditarily strictly subnumerous to $x$. (which is a theorem of ZF ([Holmes][2])).


  [1]: https://mathoverflow.net/questions/309562/can-cardinality-be-defined-with-essentially-no-practical-restriction-on-non-well?answertab=active#tab-top
  [2]: https://randall-holmes.github.io/Drafts/hereditary.pdf