After I could not get it to charge up, I returned the ASUS TF101 tablet to Staples. The tech staff there gathered around to see what was wrong with the battery. Sure enough, it showed 0% charged, with the red exclamation mark.
Looking up their online support system, one suggestion was to reinstall Android. Tried it. No change.
So they got out another unit from the back. Same problem: battery level was 0%, and it would not charge. They hauled out several more units from the stock room. Same problem: battery level was 0%. Finally, they found one that had a battery with 1%, and after plugging it into the power supply, the battery began charging.
Looking up the problem online, I found that that ASUS made a design error. When the battery drains completely, it goes dead; it cannot be revived -- except under warranty replacement, should your unit still be under warranty!
Normally, all portable devices have a chip that monitors the battery level. When the level gets low, you get a warning; when the level gets too low, the portable device is shuts down to protect the battery. ASUS seems to have left out the monitoring chip, it is speculated.