Q:

Strongest large cardinal axiom compatible with $V = L$? 

What is the strongest known natural large cardinal axiom compatible with $V = L$ (strongest in the sense that it implies all known "small" large cardinal axioms, where a large cardinal axiom is said to be "small" if it doesn't imply $V \neq L$)? 
One candidate might be "there is an $\alpha$-Erdos cardinal for every countable ordinal $\alpha$."
Using the ``instrumentalist dodge'' of Steel/Hamkins, it seems that we can "cheat" to obtain stronger large cardinal axioms within the confines of $\operatorname{ZF}+(V = L)$, such as "there is a transitive model of the theory ZFC+"$0^\sharp$ exists.""   I will leave whether or not such axioms are "natural" for a matter of debate here.   
Disclaimer: I'm an algebraist and number theorist but am currently trying to grasp the big picture in set theory, large cardinals, inner model theory, and the like. 
Related MO posts:
Is there a large-cardinal completeness theorem for $L$? 
If $0^{\sharp}$ exists, then every uncountable cardinal in $V$ is as large as it can be in $L$.
Erdős cardinals and $0^\sharp$

A:

Maybe a comment:
In the paper ``A large cardinal in the constructible universe'' Silver shows that if $\kappa\to (\alpha)^{<\aleph_0}$ for all countable $\alpha,$ then the same is true for $\kappa$ in $L$.
On the other hand, by a result of Rowbottom, $\kappa \to (\omega_1)^{<\aleph_0} $ contradicts $V=L.$
Silver concludes with the following:

It does not seem extravagant, then, to assert that, for all practical purposes,  $\kappa \to (\omega)^{<\aleph_0}$ is the strongest strong axiom of infinity know to be consistent with $V=L,$ and therein lies its chief interest. 