I applied through a friend who worked there and was contacted that same day by a recruiter. The two phone interviews were scheduled in the following weeks. Both interviewers were extremely kind and asked good questions that could be quickly be turned into heavy quant or explored more creatively as needed (example, see the Oil Tycoon question below). Overall the questions were not overly difficult, but seemed to focus more on your motivations to work for Palantir as well as multiple requests to describe complex analysis you've done in your past. They really seemed to love whenever I got technical with my answers or threw in basic statistical jargon or models to explain myself. The next day after my second phone screen I was called back by the recruiter and invited for an 'accelerated interview' schedule the following week. I was flown to their offices, put in a very nice hotel, with all expenses paid. Everyone was incredibly genuine and nice, even the other candidates I met interviewing that day. I went through five half hour interviews without the demo hour that other candidates received. Surprisingly, all of the interviews were non-technical. I had been preparing for the algorithms, logic, and statistics problems described here, but didn't have a single one. Instead the interviews became very psyco-analytical. I was repeatedly asked across interviewers to explain my motivations behind decisions, how I feel about things, what I would do in certain hypotheticals, and generally, why why why. It was a rather bizarre set of interviews, and since everyone was so nice and the questions ended up being so personal I had no idea how to judge my performance. The last interview was the founder interview. It was extremely short, lasting about 5 minutes. He asked similar questions to everyone else, Why Palantir, How did you hear about us, Why did you do X on your resume, etc. He kicked me out after a few short answers but I was told not to worry, that this happened often. It appeared he was busy thinking about something else while speaking with me anyway, and I didn't take it personal.
I got a call a week later saying that it had been a very close call, but ultimately I would not be given a position. Overall, it was a very positive experience. I'm sad about the outcome but still enthusiastic about the company and its potential.