When does $V=L$ becomes inconsistent? 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 $A$?
 A: The borderline seems to be very near the $\omega_1$-Erdős cardinals.
The Wikipedia page on $0^\sharp$ explains it thus:

The existence of $\omega_1$-Erdős cardinals implies the existence of $0^\sharp$. This is close to being best possible, because the existence of $0^\sharp$ implies that in the constructible universe there is an $\alpha$-Erdős cardinal for all countable $\alpha$, so such cardinals cannot be used to prove the existence of $0^\sharp$.

In particular, an $\omega_1$-Erdős cardinal is inconsistent with $V=L$, but having $\alpha$-Erdős cardinals for countable $\alpha$ is fine with $V=L$. All the smaller large cardinals in the hierarchy (see the Wikipedia large cardinal list) also appear to relativize down to $L$. 
A: Just a supplement to Joel's answer:
The exact boundary for existence of $0^\sharp$ was pinned down by Klaus Gloede:  $0^\sharp$ exists if and only if there is a cardinal $\kappa$ such that every constructible partition of $[\kappa]^{<\omega}$ has an uncountable homogeneous set.
Gloede, Klaus, Ordinals with partition properties and the constructible hierarchy.
Z. Math. Logik Grundlagen Math. 18 (1972), 135–164.
As Joel points out, everything below this on the usual large cardinal hierarchy seems to be compatible with L.  
