Well, looks like my recent purchase of an iPod Shuffle was somewhat prescient. My original 20GB first-generation iPod gave up the ghost, showing the "sad iPod" icon just after I recharged it:
I'll try to revive it (apparently the hard disk won't spin up, which is occasionally fixable), but it looks like I'll have a brand-new 2100MAh battery, an iTrip and a spare Firewire charger for sale...
Mental Note: Don't buy a larger-capacity iPod until it has no moving parts and a removable battery. Or until the iPod mini switches to flash storage.
Update: Turns out that it was the new battery that killed it. Upon opening it, I found that the Newer Technologies battery had inflated to the extent of pressing down on the PCB and hard disk. Pressing slightly on the case before opening it made a grinding noise (which I suppose was the disk being compressed), and restoring the original battery solved the problem (a much slimmer fit, and no more grinding noises when applying pressure to the iPod).
The iPod is now being repaired in Disk Utility, although I fear the disk itself may be beyond salvation (those grinding noises had to be something in trouble).
We'll see. It would be sad to let it go, since it's been around for ages...