Edit: the question was answered to the negative because $ZF$ proves the existence of Hartog numbers. So this calls for a modification of the question to be in just $Z-\text{Regularity}$

Is it consistent with $Z-\text{Regularity}$ [instead of $ZF-\text{Regularity}$ in the original question] to have a set that is strictly bigger in cardinality than any set in the cumulative hierarchy of $Z$?

Formally: Is it consistent to have $Z-\text{Regularity}$ plus

$\exists x\, \forall y\, \big[\exists \alpha\, (y \in V_{\alpha}) \to \exists f\, \left(f\colon y \to x \wedge\, f \text{ is injective}\right) \wedge \not \exists g\, (g\colon x \to y \wedge\, g \text{ is injective})\big] $

Where $V_\alpha$ is defined in the usual manner.

The version of $Z$ defined here have all stages of $ZF$ below $V_{\omega+{\omega}}$