Working in $\sf Z- Reg.$ we can have sets bigger than every well-founded set, let's label such sets as "remote". Now, suppose we'll add the axiom of existence of transitive closures, and the axiom stating that for every set $x$ there exists a set $\mathcal H_x$ of all sets hereditarily strictly subnumerous to $x$ and such that there exists an injection from $x$ to $\mathcal H_x$. This is compatible with the existence of remote sets. The point is that we can work in a milieu outside of regularity and choice where the notion of cardinality is definable! So, here we define a cardinal as an equivalence class of all sets hereditarily strictly subnumerous to a common set, under equivalence relation bijection. So the cardinality of $x$, denoted by "$|x|$", is the set of all ranges of injections from $x$ to $\mathcal H_x$. Now, the cardinalities of remote sets shall be called "remote cardinals".

Now, is it consistent to add an axiom stating the existence of a set of minimal remote cardinality? That is: $$ \exists x : \operatorname {remote}(x) \land \forall y \, ( \operatorname {remote}(y) \to |x| \leq |y|)$$