TITLE: Large cardinal consistency strength and size
QUESTION [12 upvotes]: My understanding is that large cardinals are ordered by "consistency strength", but how does this correlate with their size (cardinality)?
More specifically, are there any systematic results on the lines of:
If A and B are two types of large cardinals such that 
Cons(ZFC + Type A exists) => Cons( ZFC + Type B exists)
THEN
Cardinality of smallest Type A cardinal >= Cardinality of smallest Type B cardinal

REPLY [5 votes]: Since the relevant kind of large cardinal was just named on Joel Hamkins's blog, let me point to it as an interesting dual to some of the existing answer.
A cardinal $\kappa$ is otherworldly if there is some $\lambda$ such that $V_\kappa\prec V_\lambda$. (Exercise 4.1.7 in Drake's large cardinal book shows that otherworldly implies worldly, hence the name). We say $\kappa$ is totally otherworldly if there are arbitrarily large $\lambda$ such that $V_\kappa\prec V_\lambda$.
Now totally otherworldly cardinals are very low in consistency strength: a single inaccessible cardinal suffices to show the consistency of a proper class of totally otherworldlies, via the usual Löwenheim–Skolem argument. But every totally otherworldly cardinal is $\Sigma_2$-correct, which makes it at least larger than the least measurable, the least superstrong, and the least huge, if consistent.
So this is an example of very low consistency strength but very large size.