I solved the problem on my iPad. It will likely work for you as well if these are your symptoms:
-- when plugged in, the iPad goes into a loop that never gets past the Apple logo
-- between loops, you see an occasional red battery icon that indicates the iPad is off and charging
-- you can initiate recovery mode, but it always ends in some sort of error message on iTunes
Here is the issue. Your iPad has insufficient battery power to start up. Normally the iPad will not even ATTEMPT to restart until there is enough battery power to do the job. In this case, there is a software glitch that is telling the iPad to try to boot even though the battery can't last long enough to finish. So round and round you go, charging a little, attempting to boot, shutting off, charging a little .... ad infinitum.
Your goal is to get the iPad sufficiently charged to complete the boot cycle. Here's how:
-- plug the iPad into a power source. Ideally, use a 10W charger for the iPad, not the 5W iPhone charger. Likewise, ideally do not plug it into your computer either, as the Mac will only charge at 5W as well. Don't have the 10W charger? No worries, proceed below and add a little charging time to be safe.
-- find a way to hold down the home button (front of the iPad) and the off/on button (upper right of the iPad) simultaneously for at least a few hours. This prevents the loop cycle and allows the iPad to finally charge. How to hold the buttons down? Use your ingenuity. I used strong gaffers tape to hold down the on/off button and a clamp (gently applied!) with a thin sponge under it to depress the home button.
-- After a few hours or less, the iPad will be sufficiently charged to restart normally.
As for why this loop occurs to begin with, I can't say. In my case I let the iPad run down, but I had done that many other times with no issues.