Denote by $ZF^\times$ the theory of $ZF$ without the axiom of infinity. We know that $V_\omega$, the set of all hereditarily finite sets in a model of $ZF$, is a model of $ZF^\times$.
We further know that $\newcommand{Ord}{\operatorname{Ord}}\Ord^{V_\omega}=\omega$. Consider the following formula: $\varphi(x,y)=(x \text{ is an ordinal})\land (y\text{ is an ordinal})\land \Big((x\neq0\land x\in y)\lor y=0\Big)$
It is not hard to see that the class $\{\langle x,y\rangle\mid \varphi(x,y)\}$ is a well-ordering of order type $\omega+1$. Similarly we can define $\omega+2$, even $\omega+\omega$. We can go even further, much further.
However, we can obviously go so far, there are only countably many formulas and countably many parameters so there can only be countably many order types definable.
Questions:
What is the least ordinal not definable in $V_\omega$ from $ZF^\times$?
We can push this question into $NBG^\times$, defined as $ZF^\times$. Now we can quantify over classes, surely we can push this even higher?
Vaguely related: