The Santa Fe Institute is one of the most exciting places to do science, and even just to think. Here I've jotted down some notes for the prospective postdoc, and refined it with some helpful feedback from Aaron Clauset and Dan Larremore. Some underappreciated aspects of SFI:

• The core researchers at the institute are only a dozen or two postdocs and a handful of resident faculty. The rest of the institute are visitors, whose visits range from half a day to a semester.
• The fellowship is, as you've probably heard, deeply unstructured by design. The idea is to give postdocs freedom to pursue their own research agenda for three years. I've heard it said that it's more like an assistant professorship than a postdoc position. My experience of this is that it's essentially true, and the biggest constraints to doing the most daring things we can think of are our own fears: fear of failure, fear of judgment, fear of career uncertainty. To be fair, those can be real and substantial. At SFI, you will be in good company facing those fears, but nobody will force you to do it either.
• One might say that the freedom of SFI reveals the profound limitations of any potential academic career. It's like working at a company with "unlimited vacation". You still need to publish to get a job. Or you can fully stop caring, but this is precisely one of the things that the SFI interview is trying to gauge. Do you have the self-directed enthusiasm to work without someone telling you what to do? And will you actually choose a path for yourself, or are you simply chugging along on the tracks laid for you by your PhD advisor?
• The other concern that SFI folks will have, perhaps bigger than a postdoc who squanders their time, is someone who will join the institute and use their affiliation to spread nonsense. SFI is so decentralized that there's not a strong grip on how researchers communicate about the institute to the outside world. Complexity science also tends to attract weirdos, and SFI has a very delicate and intentional relationship with its own weirdness.
• You should think seriously about whether you actually want this freedom. For instance, if you decide that for your intended job market packet, you want to have certain publications in certain venues, then perhaps there is utility to joining a lab that has a track record of publishing papers in those venues. You may also want more of a dedicated mentor. A more conventional postdoc will give the added bonus of a mentor and a "guaranteed" recommendation letter, which isn't the case at SFI, unless you go out of your way to collaborate intensively with someone. To be fair to SFI, though, if you let your curiosity flow, you will find people to collaborate with!
• Because there are so many thoughtful interdisciplinary researchers passing through, the SFI postdoc is also a science communication bootcamp. I was able to practice many variations of how I introduced myself to different waves of visitors.
• SFI is very much a community (or a collection of communities), and every postdoc leaves a deep imprint on the culture. People are on the lookout for kindness and good scientific citizenship, as well as a sense of reciprocity. SFI tends to like people who can contribute to it AND who would benefit and grow from the association.

Some more specific words of advice about the SFI postdoc application: (Note: this has changed since I left SFI, so the details of how the selection happens may be out of date. Nevertheless the same social dynamics probably apply.)

• The application process has traditionally consisted of several rounds, where a small committee whittles down the (giant) pool to 20–40 people that all of the postdoc and faculty at the institute evaluates together.
• It's a ritual that is taken very seriously at SFI. Even if many people form opinions before the meetings, people can and do get persuaded by others. Everyone is allowed to speak, and everybody has the same weight of vote, whether postdoc or faculty. However, the faculty have gone through many rounds of this rodeo, and can be very persuasive.
• What's especially damning is if your application reaches out for someone's support, but that person rejects you. For example, if you write in your application that you would really like to collaborate with person X, but person X says to the room that they have no interest in working with you — that's very bad for your case.
• Don't write some fluffy thing about how complexity is the meaning of life for you.