I was first contacted by a recruiter on linkedin in September 2015 asking if I was interested in applying. I had a quick phone screen with that recruiter shortly thereafter, and was put in touch with a recruiter in a different department more aligned with my interests shortly thereafter. This 2nd phone screen was fairly difficult- this particular recruiter had an engineering degree, and really grilled me on the technical aspects of a few of my projects. I'm not sure if this is normal for recruiters at SpaceX, but keep in mind that this phone screen may not necessarily be an easy one.
I then had a phone interview with an engineer, who asked me a ton of technical questions about my project, as well as a few general technical questions and a brain teaser (all outlined below). After that interview, I didn't hear back from SpaceX at all for 4 months. I emailed and called my recruiter multiple times, but never heard back. I assumed that they were not moving forward with my application, and continued my job search, applying with other companies.
You can imagine my surprise when a 3rd recruiter called me in late January 2016, asking if I would be interested in "re-engaging" with SpaceX. I was set up for a phone interview with a 2nd, more senior, engineer, who asked more technical questions about my project. He was very apologetic about it taking them so long to get back to me. It was a really interesting discussion, and was a very enjoyable interview on the whole- really forced me to rethink certain aspects of my project from the perspective of a rocketry engineer. A few days later, I was contacted asking me to come out for the on-site interview.
When I got to HQ in Hawthorne, I got the tour of the facility and then gave a presentation to ~8 engineers on a technical project I had worked on. It had to be 15 minutes long, but took about an hour with all of their questions. After the presentation, I was taken to a separate room to wait while they debated on whether or not I should move on to the next stage of the interview process (one-on-one interviews with each of them).
In the end, I did not make it any farther in the process. It's hard to say why exactly, I thought I nailed the presentation and answered every single one of their questions accurately and succinctly. All my recruiter said was that they didn't think I'd be a good fit on their team, though she also mentioned I could try again when I had more experience. That would make sense considering I'm currently entry-level, though I do have multiple internships under my belt.