Interview process began via a referral from within the company. The process took ~ 8 weeks (though this was not Palantir's fault - one of the primary people they wanted me to interview with had a family situation). The position I applied for is located in McLean, Virginia.
The first interview was a "sanity check" call from a recruiter. He, and everyone else I spoke with through the process, was friendly, welcoming, and well-prepared. Questions were related to my background and my desire to join the company; he provided me with good information about the specific position as well as the overall company structure. This call lasted nearly an hour and was very conversational.
The second interview was a call with a Palantir employee who is actually doing the job I was applying for. This was a more technical interview, though also very friendly and engaging. More information specific to the role and some moderately challenging (for me) administrative questions (e.g. describe the RAID levels and in what kinds of scenarios you would use each). This call again lasted around an hour.
The third interview was another phone call, but this time with a Site Reliability Lead. The Systems Analysts work very closely with others on the Site Reliability team (especially the engineers), so this interview was more about my skills with customer/client interaction, project management, organization, and teamwork. It was again a friendly and conversational interview, and again lasted around an hour.
After the third interview I was asked to come on-site, and the company paid for my trip to McLean (flights, hotel, $40/meal, car service). I was put up in a very nice hotel about 5 minutes' walk from the Palantir office (there was also a complimentary shuttle). The night prior to my trip, I had a brief call with my recruiter where he gave me some "interview prep" by explaining who I would be meeting with on-site and some basics about what each one of them would ask. He specifically told me that there would be "no brain teasers" and he was correct.
The on-site interviews consisted of meeting with 5 different people over the course of about 4 hours in a couple different rooms (including one employee's office). I spoke with one employee over a video conference, but the rest were in person. Just as in the phone interviews, the people I met with were extremely friendly, seemed genuinely interested in me not only as a potential employee but as a person. They shared their own stories about how they became attached to the company, and I did not get any of the "cool aid" mentality that I've read about on this site and elsewhere. They just really seemed each to love their job.
The interviews were a pretty good mix. I got some technical questions, lots of questions about why I wanted to leave my current position and why I specifically chose Palantir, lots of "fit" questions (they are very concerned with hiring the right personality). The people I spoke with were in several different roles, and included one of the guys I had interviewed with over the phone. By the end of the interviews, we were talking about things like "do you think you would be able to do X on your first day?".
I did not have an interview with the founder as many others seem to have had. I believe that, since the company has expanded so rapidly, candidates outside of Palo Alto probably don't always have the chance of meeting with him. Or, hey, maybe it's a bad sign. :)
All in all, this was a very enjoyable, though lengthy, process. I got the impression that, more than anything else, this company is looking for very talented people, even if their talents might be very specific in scope. I'm choosing "No" for "did you receive an offer" because I'm still waiting on an official decision from Palantir. [UPDATE: received official "No Offer" response from Palantir]
This company is growing hand over fist while still trying to maintain a "start-up" feel, and, based on my experience, they are doing a pretty good job.