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In light of the celebrated theorem of Vop\v{e}nka (1972), proving in ZFC thatevery set is generic over HOD, it is natural to inquire whether theset-theoretic universe $V$ must be a class-forcing extension of HOD by somepossibly proper-class forcing notion in HOD. We show, negatively, that if ZFCis consistent, then there is a model of ZFC that is not a class-forcingextension of its HOD for any class forcing notion definable in HOD and withdefinable forcing relations there (allowing parameters). Meanwhile, S. Friedman(2012) showed, positively, that if one augments HOD with a certain ZFC-amenableclass $A$, definable in $V$, then the set-theoretic universe $V$ is aclass-forcing extension of the expanded structure$\langle\text{HOD},\in,A\rangle$. Our result shows that this augmentationprocess can be necessary. The same example shows that $V$ is not necessarily aclass-forcing extension of the mantle, and the method provides counterexamplesto the intermediate model property, namely, a class-forcing extension$V\subseteq W\subseteq V[G]$ with an intermediate transitive inner model $W$that is neither a class-forcing extension of $V$ nor a ground model of $V[G]$by any definable class forcing notion with definable forcing relations.

The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD