I installed a new battery, and it started up right away and drove fine.
Day 2: The problem came back. Here's what I've done:
- Checked the voltage on the battery: 12.6
- Turned on accessories without starting the car: no problems
- Scanned for codes: no codes
- Jump-start worked
- Checked the voltage on the (running car) battery: 14.2
- Disconnected the black terminal on the battery: car continued to run
Day 3:
I went out to the car to check on things, and it was still clicking from the night before.