I applied through college or university. The process took 2 weeks. I interviewed at Jane Street
Interview
On-campus presentation, online application. Found out through email to sign up for on-campus first round interview. I did not progress past this stage, though I have heard that if the first round goes well, you proceed to at least two more rounds. Questions were all mental math and probability and became harder with each subsequent question.
Interview questions [1]
Question 1
Play a game where you roll a die: what's the expected value? What if you can roll twice and either keep first roll or have to keep second roll? What about three rolls or infinite rolls?
They tend to examine the logical thinkers that can communicate clearly and concisely, and some problem-solving exercises intended to explore how you gather and apply new information, very different from other tech companies' interviews.
Initial application and online assessment. Had a first round behavioral interview where they asked me about myself, my experiences, and a couple of brain teasers. The interviewer was very nice.
Phone interviews (at least 3) - questions were of mixed difficulty (some pretty hard) but the interviewers were friendly and they gave helpful hints. Kind of fun - review probability and game-type questions.