I will present some recent results about new large cardinals that imply the set-theoretic universe is not HOD (the universe of Hereditarily Ordinal Definable sets). In particular, ultraexact cardinals are equiconsistent, modulo ZFC, with the existence of an I0 embedding, yet they imply that some successor cardinal is measurable in HOD. Moreover, the existence of an ultraexact cardinal together with an extendible cardinal below it is consistent with ZFC and refute Woodins HOD Conjecture. This is joint work with Juan Aguilera, Gabriel Goldberg, and Philipp Lucke.

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