(This was a reply to John that grew beyond the alloted limit.)

large cardinals have consequences that were initially surprising to set theorists.

But, John, this is simply because our intuitions about large cardinals took a bit to form, just as one would expect with any relatively recent theory. Now we understand significantly better how far from $L$ the universe ought to be in the presence of large cardinals, and find the puzzlement at Scott's result curious. We know, for example, that much much much less than a measurable already contradicts $V=L$.

Similarly, when our understanding of large cardinals was relatively poor (early 80s), it was the common belief that for any large cardinal there was a nice inner model with a $\Delta^1_3$ well-ordering of ${\mathbb R}$, and it was by refuting this that we eventually arrived at Woodin cardinals and our current view of the set theoretic universe, where (as Simon mentioned) large cardinals make such well-orderings impossible, since all projective sets are measurable.

(This significant shift in understanding started with the Martin's Maximum paper by Foreman-Magidor-Shelah, Annals of Mathematics, 127 (1988), 1-47, and is very nicely explained in 'Iteration trees' by Martin-Steel, Journal of the American Mathematical Society, 7(1):1–73, 1994.)

This also led, by the way, to the proof that large cardinals imply determinacy in nice inner models, again shifting the prior poorly developed intuitions that had us expect much larger consistency strength for determinacy than it turned out to be the case.