[Nexus 7 Q&A, Help & Troubleshooting] bftb0:
"The tablet will not shut off..." <<---- This suggests that you are seeing or observing something happening. But based on your previous posts I can not tell exactly what you are now observing - the black & white Google logo?

I would start first with trying to determine if the battery is capable of being charged. The last thing you want to be doing is fooling around with this stuff with a nearly dead battery.

If you put it on a 2A wall charger (*not* a PC USB port), does the tablet warm up a little after 20-30 minutes? If so, leave it on the charger for at least 2 hours. (Heck, even if it doesn't warm up, same advice - put it on the wall charger for at least 2 hours)

OK, next item for you to consider. In Android devices, there is usually a nonvolatile storage area (maybe in the "misc" partition?) called the BCB (Boot Communication Block). The bootloader will always look at that area first, and if there are no boot instructions in there, by default the bootloader will attempt to load & jump to the entry point of the contents of the LNX partition (Linux boot partition). This would cause a "normal" Android boot to occur by default when turning the power on. The other two common instructions in the BCB are for a reboot into the recovery, or instructions for the bootloader to stay resident but drop into it's fastboot mode.

Why am I telling you this? Because there are certain pathological situations that can develop into loops without apparent escapes. Imagine that the tablet was sitting in custom recovery mode, and the dummy behind the PC keyboard flashes some random garbage to the recovery partition, and then uses the touch interface to reboot into recovery. What happens is that "boot to recovery" instructions are placed in the BCB, and a soft-reset of the device occurs... which (when the hardware finishes resetting) loads up the bootloader.**

Now the bootloader sees the "boot to recovery" instruction in the BCB, and attempts to load the garbage in the recovery partition. Eventually - either due to a watchdog timer reset, or a illegal instruction trap, the device will soft-reset again... WITHOUT EVER CLEARING THE CONTENTS OF THE BCB.

In this example, the bootloader is intact and operational, but the looping behavior continues until the battery drains. You don't get an opportunity to communicate with it in fastboot mode because it is always trying to load the bad recovery. Either there is a way to break out of this death spiral... or their isn't.

Now, one more thing - I lied slightly right around the ** annotation.

On the Tegra3 SOC, there is a kind of micro-kernel inside the Tegra3 that is responsible for loading the bootloader. Simpler processors of old would simply load a memory page at a specific hardware memory address when they reset, but this is far more sophisticated: the little micro-kernel in the Tegra3 can evaluate the bootloader to see if it has the correct cryptographic signature and so forth.

If that micro-kernel doesn't like what it finds, the Tegra3 will put the USB port into APX mode. This is important because it is diagnostic: if your tablet always goes into APX mode, for instance after disconnecting/reconnecting the (charged) battery and then powering up the device, then you are probably out of luck; but if it doesn't go into APX mode, that suggests that the bootloader is actually healthy, but perhaps thrashing in one of those nasty boot-loop cycles.

OK, still with me?

Have you looked at this thread carefully?

http://xdaforums.com/showthread.php?t=2133986

It suggests that possibly you can get the bootloader to ignore the contents of the BCB by a carefully-timed press of the Vol-Down button - which, if it works - should let you escape to fastboot mode. This is documented in the above thread as "bootloader interrupt".

So, questions for you:

1) Are you sure the battery is well charged? (Not easy to tell without a voltmeter or heating of the tablet during charging - the temperature of the tablet will drop after it finishes charging. One way to be sure of things is to disconnect the battery, reconnect it, and then put it on the wall charger. That way you can be sure that the tablet is not on, so any heat would be due to battery charging, not other components running. FWIW, a discharged battery will be about 3.5v, and charged about 4.15v.)

2) Does the tablet always go into APX mode when it starts up, even if you only press the power button?

3) Once the battery is charged, what are you seeing on the screen during start-up? After you have disconnected the battery and reconnected it, do you see anything on screen when you put the tablet on the charger?