As a follow up to this thread, I'll report that my 1DIII LP-E4 battery has returned to normal operation after several weeks of calling for calibration with each charge. After the third (or fourth?) calibration cycle, it shows battery efficiency at maximum and enables as many shots per charge as when new (a lot!).
I know that the battery is aging and won't last forever, but I think it's worth noting that one needn't give up on a battery that calls for calibration a few times in a row.
Mine is 18-19 months old and I calibrated it yesterday but still have the message.
I will try it a few more times over the next week and report if the message goes away.