I've suggested elsewhere that this sense of extraordinariness when faced with a result like H^200 is the result of a kind of hindsight bias.

Roughly speaking, the idea is that certain notions seem simple to our brains... easier to access and understand and express and so forth. When such an notion is suggested to us, we frequently come to believe that "we knew it all along."

A H^200 string of coin-flips is just such a notion; it seems simpler than a (HHTTTHHTHHT)^20 string, for example. So when faced with H^200 we have a stronger sense of having predicted it, or at least that we would have been able to predict it if we'd thought to.

But, of course, predicting the result of 200 coin flips is extremely unlikely, and we know that. So when faced with H^200 we have a much stronger sense of having experienced something extremely unlikely (aka extraordinary) than when faced with a more "random-seeming" string.