If an extension of ZF is not consistent with $V=HOD$, does this imply that it must prove existence of large cardinals? Suppose we have a first order theory $\text{T}$ such that 1. $\text{T} \supset \text{ZF}$ 2. $\forall M [(M\models \text{T}) \to \neg (M\models \forall x (x \in HOD))]$ Does that mean that $\text{T}$ must prove existence of a large cardinal? If Yes, then among the list of the known large cardinal proeprties , which one is the least that $\text{T}$ must prove its existence?

This is just asking if $V\
e HOD$ implies any large cardinal properties. If we start with a model with no inaccessibles and then force $V\
e HOD$ (by, say, adding a Cohen real), the resulting outer model has no inaccessibles either (inaccessibility is absolute that way). Thus $V\
e HOD$ does not imply the existence of inaccessibles.