Can't go fullscreen in any app
Tried a couple more things, and here's what worked: kill the Dock, then get it to restart.  I did this by doing killall Dock in the Terminal.  It didn't restart by itself, so e.g. Cmd-Tab didn't work to switch applications, my desktop had no background, and (of course) there was no Dock.

I launched the Dock again by going into System Preferences &amp; clicking on the Dock pane.

Why did I try killing the Dock?  Because I also noticed that Exposé/Mission Control wasn't working either.  Some searching led me to this question, which suggested restarting the Dock.