Update 2017-10-07: Google Maps was the culprit. After uninstalling and reinstalling Google Maps I suddenly had gigabytes of free space.
There's something really wrong with my iPhone's software. For a couple weeks now, it has been constantly nagging me that I'm out of space. Initially, I handled this by backing up my photos and deleting those. Then it reminded me again, so I deleted my podcasts. Then it reminded me again, so I deleted a couple large applications. And now, it's still reminding me that it's out of space!
This morning I didn't believe it anymore. I had no photos, podcasts, or music on my phone, and the largest app on my phone was only a couple hundred megabytes. So I decided to add up everything that the phone said was taking up space. My phone indicates I'm using 11.64 GB of space right now. I went down the list of applications on my phone and added them up. I got a grand total of only 3.6 GB.
I guess I could take this in to an Apple store to get it fixed, but who has time for that? I'm probably just going to do a full backup and restore in the next couple days unless someone gives me a good idea about how to fix it.
Have you ever had a problem like this? Let me know about it in the comments!