WordPress prompted me to update to the latest version, and of course I figured nothing would go wrong. The update ran well enough, but I was totally unable to access the back-end of the website afterward. The process was running out of memory. I discovered that my host, NearlyFreeSpeech.net limit the amount of memory for each PHP process to 32MB unless you pay an extra fee. It's not much to get it bumped up to 64MB, only a few cents a day, but I always prefer the cheaper option. I was able to regain access to the admin panel by renaming the folder /wp-content/plugins to /wp-content/plugins.old, which prevented them from being loaded.
In the mean time I also went ahead and installed W3 Total Cache and, in doing so, ran headlong into another problem. It seems the 'tmp' folder created by the installation script does not belong to the same user as my FTP account, so even with permissions set to crazy (777) I was unable to save any settings. I had enough privileges to rename it and create a new 'tmp' folder, which ironed out the problem. Hopefully the site is running a little faster for everyone as a result.