This is a trip. The tech had me open up the laptop and pull out, then re-seat the memory chips. Then he had me take the battery out (okay, it was already out), unplug the laptop from the wall, then hold down the power button for two and one-half minutes. You read that right, 2 1/2 minutes. After that, I plugged the laptop back into the wall (not a power strip, mind. It had to be the wall.) Leaving the battery out, it powered up.
This is not to say that the laptop is running perfectly. I can still get the complete no-start problem every now and then (and that is after just one evening's testing!) I also still get the intermittent system lock-up problem that I've had before. Which, unfortunately, means that I probably should send it in and get them to look at it while it's still under warrantee. But at least I was able to run a backup of all the stuff that my last backup missed.
I got to bed kind of late, since I was frantically running backups of everything I could possibly think of while the laptop was still up. Now I'm a zombie with a headache. I wish it were Saturday, and I could pretend that this whole "being awake" thing was just a bad dream, roll over, and go back to sleep.