Well, shit. My iPhone's dead.
I noticed it around 6:30 this evening when I went to check the time on my phone (unusual, but I had been doing dishes and food prep in the kitchen and wasn't wearing a watch). Pushed the button, got no response. Pushed the other button, nothing. Held the button down, uh-oh.
I went to Apple's support site and got into a chat with a service rep, who walked me through some steps that might have gotten it going again (pro tip: holding down both the power and home buttons is the iPhone's version of a forced restart). That ended up getting it stuck in a loop, so I was told to connect it to iTunes for a memory wipe (I had backed up through iCloud) but that didn't work either.
Since I've had the phone only two months, it's still covered under the warranty, so I have an appointment at the Apple store later this morning to get a new one. The last thing I did earlier in the afternoon was update two apps, but I can only remember one of them (GasBuddy). One of the apps may have been the culprit, or it could have been something else. I'll probably never know for sure.