Just had a strange one. I got out of the shower and my battery was pretty low but not in the red zone, as I hadn't plugged it in all day. I woke the phone to see if I had any messages, and when I didn't, I put the phone to sleep again. Then a few seconds later it rebooted to the Google screen and lingered there for far too long (at least 45 seconds, maybe more?). I got scared and pulled the battery. Decided that if any recovery would have to happen, I'd put my spare charged battery in so I don't have to worry about charge.
Started up the phone again, and it lingered on the Google screen longer than usual but did successfully boot.
I tried going in with adb to a root shell and typing cat /proc/last_kmsg > /sdcard/last_kmsg.txt
but I just got: sh: cat: /proc/last_kmsg: No such file or directory
So I guess there is no last kmsg.