I applied twice. The first time, I applied through the Blizzard Careers web site, and was rejected via an automatic response two weeks later. The second time I applied through employee referral and was able to go through the whole process. About two weeks after I applied, I heard back from a coordinator. We set up an hour-long phone interview with the team lead and the engineering manager. A day after the phone call, I received an online take-home test. A week after I finished my take-home test, we scheduled an on-site interview for a week later (so from first contact to the final on-site interview, it took about 4 weeks). After the final interview, I received frequent updates from the recruiter regarding the status of the role I was applying for, and was given an offer 3 weeks later.
The interviews were difficult but manageable. I was especially nervous due to how much I wanted to work at Blizzard but my interviewers did a great job of making sure I was comfortable. Make sure you review computer science/software engineering/C++ (or the programming language of your team's choice) fundamentals and be familiar with everything posted on the job listing itself. For me, prior industry experience helped a lot. Also, from beginning to end, you should expect a lot of personality and behavioral questions. You should also make sure you can talk a little bit about everything you list on your resume, and be prepared with questions for the team, their culture, and their processes. This may be unfamiliar territory for software engineers, but something that Blizzard places a lot of emphasis on is your personality and whether you'd be a good team player/cultural fit. This was evident in all of my interactions with folks during the interview process--I even had great conversations about games with the receptionist while waiting for my interview to start. An employee who was just passing by the front desk also wished me good luck (thanks!).
In sum, I really enjoyed my experience. During my on-site, I felt like I had a taste of Blizzard company culture as well as the culture of the team I applied to. It was quite different from my previous interview experiences at other game studios and software companies, most of which felt very technical and devoid of culture. I would definitely encourage people to err on the side of applying. I know that personally I psyched myself out of applying to Blizzard for multiple years. It never hurts to apply. Even if you fail, you can always try again later. But if you don't try you may be haunted by a dull sense of regret knowing things could have been different :) Having an internal referral helps a ton for the visibility of your application, so make sure you network!