Reinhard cardinals are the largest cardinals yet defined that are consistent with ZF. This suggests that there are large cardinals that are inconsistent with ZF - are there in fact any?
So, there is a very facetious answer to your question; it depends what you mean by large cardinal axiom.
For instance 0-sharp is something like a large cardinal axiom without asserting the existence of a cardinal explicitly. It's rather about sets of indiscernibles that code the construction of L, and is very naturally viewed as a (Delta^1)_3 non-constructible real. However, 0-sharp fits very nicely into the large cardinal hierarchy, so is something like a large cardinal axiom; it's a principle of strength.
If we're talking about principles of strength then, there is a very obvious axiom that is inconsistent with ZF, namely 0=1. However, this is obviously not what you're looking for, rather you want an axiom that is in some sense "natural" or "conceivable" or "contentful" (or some other similarly philosophically problematic notion; cans of worms everywhere!), but that is inconsistent with ZF. Hugh Woodin and Peter Koellner have been working on thinking up such a large cardinal notion inconsistent with ZF; the hierarchy is roughly outlined in the Wikipedia article you cite (I would be somewhat careful of that, however; there are a couple of errors in the text). No inconsistency has yet been found, however.