Assume that $V\neq HOD$ and let $\kappa = \min \{\alpha\in On \mid \mathcal{P}(\alpha) \not\subseteq HOD\}$.

Clearly, $\kappa$ is a cardinal.

**Question:** Is it consistent that $\kappa = \aleph_\omega$?

Note that it is consistent that $\kappa$ is a regular cardinal: start with $V=L$ and force with $Add(\kappa,1)$. Since this forcing is weakly homogeneous, its generic filter is not in $HOD$. Since we don't add any bounded subsets to $\kappa$, for every $\alpha < \kappa$, $\mathcal{P}(\alpha) \subseteq L \subseteq HOD$.

Similarly, it is consistent that $\kappa$ is singular with countable cofinality. Let $\kappa$ be a measurable cardinal and let $V = L[\mu]$ ($\mu$ is a normal measure for $\kappa$), the canonical inner model for one measurable cardinal. Let $C$ be a Prikry sequence. Then $HOD^{V[C]}\cap \kappa^{<\kappa} = L[\mu]\cap \kappa^{<\kappa}\subseteq HOD$, but since the Prikry forcing is weakly homogeneous, $C\notin HOD^{V[C]}$.

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