This question is anonymous for obvious reasons.
I referee what feels like a decent number of papers (though I don't know how many is normal!), and I try to take it seriously. Sometimes, based on something I explicitly said or something implicit in my review, the journal rejects the submitted paper. Usually this seems to be because the editor feels that the paper isn't up to the journal's standards.
In an event that has proven less rare than I would like, I am sometimes asked - by a different editor and journal entirely - to review the same paper again. My record is currently three such rotations.
Usually I try to beg off, because I feel like the author deserves to get a new referee who is less biased and might bring new insight to their review. What I've found in practice is that editors tend to persist, and solicit my opinion more informally. I understand this, because I know that referees are hard to find, and so I've usually done so. The editor usually will say that they will seek out an additional referee, though I rarely hear the final story. In some instances I have gotten the impression, for reasons that I don't want to get into, that an author is starting to feel persecuted due to repeated rejections by prestigious journals.
What is the best policy on requests to repeat a refereeing job?