As much as a source of documentation for myself as for you people out there that may or may not exist on the other side of the screen:
Over the past day or two, I've noticed that my iPod (60GB, w/Video) was sitting at the boot screen (the one with the chrome apple on a black background) while plugged in to the Mac. Looking inside iTunes, I noticed that the iPod wasn't recognized as being connected. I tried resetting the device a few times (toggle the hold button on top on and off, followed by holding the "MENU" part of the click wheel while simultaneously holding the center button), which would drop me to some monochrome version of the boot screen, followed by a monochrome charging display, followed about a minute later by the boot screen again.
Googling around, I found http://www.ipodbatteryfaq.com/ which told me a little bit I already knew.
It then led me here: http://www.apple.com/support/
That took me through pretty much the same steps that I had been doing, so I kept clicking on the "No, this step didn't solve the problem" links on the bottom of the page. The last, most terrifying step was to "restore" my iPod, which told me to connect my iPod up, and then iTunes would recognize it. No, sorry - that's the problem. If I could walk that way, I wouldn't NEED the talcum powder.
Clicking at random, I came across this page: http://docs.info.apple.com/article.html?artnum=93936
This identified that my battery was very low (which I sort of suspected from the monochrome graphics, also the fact that the graphics indicated that it was charging), and there was a suggestion to try charging from a wall adaptor. I'm not sure if I even have a dedicated power cable for that device, but after a while I recalled that I have an adaptor I got for my iPod shuffle that has a USB socket on one side and an AC plug on the other. Just the thing for charging a USB device.
Leaving it plugged into the wall for a half hour or so seemed to be enough to get it bootstrapped back into a mode where it would reach the menu. I proceeded to plug it back into the Mac, where it was happy to sync up again, and it seemed to be charging happily, as well.
I'm a little perplexed that I could discharge the thing so far that it couldn't be charged by a USB connection to the Mac, but it seems to be working again, so as long as I can remember to try charging it from the wall, I should be OK. And I was worried that I might get on a plane in 8 days and not have music to accompany me. Or - practically as bad - only have 1 GB of music. How did people live in those times?