I've moved this thread to the Galaxy Nexus forum to help see if a solution can be found.

I'd like some more information. What version of the Gnex are you using (Verizon, Sprint, or GSM)? Are you rooted and/or running a custom rom or kernel? If so, have you messed with over/under clocking and/or volting?

 

Does the phone boot up all the way or does it shut off in the middle of booting? If you put the phone in fastboot mode (by holding the two volume buttons and then holding the power button while off), does it shut off quickly or will it remain in that mode?

 

Alright, that is good news. This means that it's software and not hardware. This should easily be solved by a simple factory reset. Did you ever unlock your bootloader by any chance?

 

Alright, so if the factory reset in recovery didn't help, then the next thing we'll try to do is to unlock the bootloader. This give the device a complete wipe of everything. If this doesn't fix it, then we'll try to flash the factory image. The easiest way to do this is through Wug's Nexus toolkit if you have windows. You can download that here: WugFresh

Download and install that. After it is installed, you'll want to click the Install drivers option, and use option 2(signed drivers). Follow the tutorial for installation, but some steps are going to have to be skipped since the device won't stay booted up. We're going to set up the fastboot drivers so that this can be done. Do that for now, and when you are done or run into an issue, let me know and we'll go from there.

 

have you possibly tried a different battery in this nexus?

if you get the same problem it is not software related, and it is a device issue.
i dont see unlocking the bootloader as the fix for this issue. but it is your device and you can try whatever you want with it. i would first try a different battery before i begin to unlock my phone, especially if you do not want it locked in the first place. because once it's unlocked you cannot return it that way. you would need to lock it back.

 

I'm pretty convinced that it's software because the device doesn't shut down while in recovery or fastboot. That to me says it's software or kernel based. Granted it could just be coincidence that it stays on in those modes.

Testing an extra battery would be a nice control to see for sure though.