= Solution
The <cumulative hierarchy> is defined by <transfinite recursion>:
$$
\boxed{V_0=\varnothing,\qquad V_{\alpha+1}=\mathcal P(V_\alpha),\qquad
V_\lambda=\bigcup_{\alpha<\lambda}V_\alpha\quad(\lambda\text{ a limit ordinal}).}
$$
This recursion on <ordinals> is available without assuming the <Axiom of foundation> for all sets. The stages are increasing and transitive, by <transfinite induction> using their definitions.
Work internally in the specified model of ZF without the <Axiom of foundation>, and assume the <membership induction theorem>. Apply it to
$$
\varphi(x)\equiv\exists\alpha\in\mathrm{On}\ (x\in V_\alpha).
$$
Suppose every member $y$ of $x$ lies in some stage. Assign to each such $y$ the least <ordinal> $\alpha_y$ for which $y\in V_{\alpha_y}$. The <Axiom schema of replacement> gives a set of these <ordinals>, and let $\beta$ be their supremum, taking $\beta=0$ for the empty collection. Monotonicity gives $x\subseteq V_\beta$, and therefore $x\in V_{\beta+1}$.
This proves progressivity of $\varphi$, so membership induction gives it for every set of the model. Every stage already consists of sets in that model. Consequently
$$
\boxed{V^*=\bigcup_{\alpha\in\mathrm{On}}V_\alpha.}
$$
The equality is internal to the given model; its <ordinals> and power sets are computed there.
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