Throughout this process I got the sense that Stripe wanted to assess my rapport with various different team members and get multiple people's perspectives on my candidacy.
1) Phone screen -- basic questions about your background and thoughts on the company. One hypothetical problem to respond to.
2) Writing exercise, answering hypothetical questions from users.
3) Phone screen #2, same general territory but slightly longer and with a different person.
4) Onsite interview with various members of User Ops (none of the ones I'd talked to earlier or corresponded with). This consisted of a) informal lunch, b) general interview, c) simple debugging exercise, d) timed writing exercise, and e) interview with the head of North American User Ops.
My impressions: You need to be friendly and good at communicating (always true but especially so), have a logical mind, and preferably know some code beforehand. Just a little bit.