  In light of the celebrated theorem of Vop\v{e}nka (1972), proving in ZFC that
every set is generic over HOD, it is natural to inquire whether the
set-theoretic universe $V$ must be a class-forcing extension of HOD by some
possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC
is consistent, then there is a model of ZFC that is not a class-forcing
extension of its HOD for any class forcing notion definable in HOD and with
definable forcing relations there (allowing parameters). Meanwhile, S. Friedman
(2012) showed, positively, that if one augments HOD with a certain ZFC-amenable
class $A$, definable in $V$, then the set-theoretic universe $V$ is a
class-forcing extension of the expanded structure
$\langle\text{HOD},\in,A\rangle$. Our result shows that this augmentation
process can be necessary. The same example shows that $V$ is not necessarily a
class-forcing extension of the mantle, and the method provides counterexamples
to 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.
