My point is that it would limit it regardless. So what's the benefit? Is there a gain elsewhere, perhaps in the password cracking methodology?
Yes. While it's true that the rule limits overall entropy, I think the real question you should ask is how common passwords with three identical characters in a row really are.
Here's an experiment you could do: Get a list of common passwords (ten thousand most likely ones, for example). Count the number of passwords that contain three identical characters in a sequence.
Then calculate how often such a sequence should occur if it was distributed randomly. If there are more passwords with such a sequence than statistics suggest there should be, then people tend to prefer such passwords, which means that it makes sense to forbid such patterns, because password guessers would capitalize on this knowledge and try passwords with these repetitions first, which would greatly reduce the search space.
I haven't done the math, but I have a very strong gut feeling that the reduction in password entropy is the lesser problem IFF the identical-character-pattern is actually common in passwords.
If someone can calculate that for me and show me the steps, that would be greatly appreciated.
I'm too rusty to come up with a mathematically sound answer, but when I was still in school we calculated probabilities of drawing a number of black balls from a sack containing black and white balls, with putting the balls back. I think that would probably be the way to do the math correctly, and I think that since it's textbook probability, you might get lucky googling for this kind of problem.
For a quick feeling for the number of possibilities you'd remove with the not-three-identical-characters-in-a-row rule, think about what the chances would be to roll same number of eyes with a die three times in a row (the first one doesn't matter, but the following two throws each have a 1/6 chance, so 1/36). If you do the same with a password and assuming it's got 64 unique characters to choose from, you'd end up with a chance of 98.98% NOT to get an identical 3-character-sequence (1-(1/64)^2). However, this isn't correct yet because your password isn't just 3 characters long, it's 10 characters, so you'd have to take that into account. Possibly you have to multiply the chance to hit a three-duplicates-sequence (1/64 * 1/64) by 8 because there's 8 possible positions the sequence can be found in (which would still leave 99.8% of the original entropy)
Edit: Replaced math of questionable use with more sound math.