I have had the Prime for about a week now and just got back from returning it to Best Buy. I saw a problem where I could not boot when running on battery. There was also a graphic glitch (a twitch of the Asus screen during the boot process), and then about 10 seconds after that, the revolving circle which shows that the unit is booting just stopped. After that, I've seen different things happen. Sometimes, when just leaving the tablet alone, the screen goes dark about a minute after the boot process icon stops, and the tablet reboots after an additional period of time. Then the same thing happens again - the glitch and the boot process stops. Other times, I could do a long press on the power button, get the haptic feedback - then nothing - then I would do another long press on the power button, get the haptic feedback, and the boot process starts. I did eventually successfully boot from battery, but it took 4 tries going through the boot, stop boot, power button, power button, boot, stop boot...
When I boot with the unit on the charger, it will boot every time, and when the tablet is booted with the charger plugged in, there are no graphic glitches visible while booting.
Then I started having the tab lock up after being on for a minute or two after booting. This started happening today. I reset the tablet to bring it back to the factory settings (using the Factory Data Reset in the Privacy settings) and tested the boot again without the charger. It worked once, and then it locked up after approx. one minute. The power button had no effect and the screen would not respond to touch.
I'm going back to my Xoom for now, but will order another Prime soon. I just think I'll give Asus some time to work out the bugs before I order another one.
Oh, one other comment - I also had a problem with the charger plug that connects to the Prime - the cap for the connector came loose. I'm sure it can be glued - but really, I only had it a week and plugged/unplugged the charger into the tab maybe 15-20 times. It needs to be much more robust than that.