I would guess most people become famous before they realize the advantage of anonymity, and then it's too late to start with a fresh name.
But it's also possible that it's simply not worth the effort, because when you become famous enough, someone will dox you anyway.
Could be interesting to know how much advantage (trivial inconvenience for wannabe stalkers) provides a pseudonym when your real name can be easily found on wikipedia; e.g. "Madonna". Or how big emotional difference for a potential stalker does it make whether a famous blogger displays their photo on their blog or not.
My favorite anonymous person is B. Traven.
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