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 $L$.

baby mouse, this is a structure $J_\tau$ with an external measure that is iterable (and some technical definability requirements). Asking that for each $\alpha<\omega_1$ there is such a structure, but only $\alpha$-iterable, seems as strong as possible. But this is in essence the same as considering $\alpha$-Erdős cardinals. $\endgroup$ – Andrés E. Caicedo Dec 13 '14 at 1:23