In a wonderful course I'm taking with Magidor we are finishing the proof of the Covering Theorem for $L$.
The theorem, in a nutshell, says that $V$ is very close to being $L$ if and only if $0^\#$ does not exist.
It is consistent that no large cardinals exist and $V\neq L$, and it is consistent that there are inaccessible, Mahlo, weakly-compact, ineffable and so on and yet $V=L$.
On the other hand, if there exists a measurable cardinal then $V\neq L$, as Scott tells us, and in fact a Ramsey cardinal is enough for that. This way we slowly fine tune the demands until we reach the definition of $0^\#$.
My question is, basically, if there is a weaker notion than $0^\#$ in terms of consistency of large cardinal axioms, which is inconsistent with $V=L$?
I am aware of the theorem that a supercompact implies $\forall A(V\neq L[A])$. Are there any other ways to measure how far are we from $L$ or $HOD$ or even $HOD[A]$ for some or all $A$?