I am using an ATmega328p-AU. I am powering it with a 2 AA battery pack and using the 6 pin ISP header. The target power option is disconnected.

The target board I'm using is just a breakout board that's plugged into a breadboard. I made the breakout board myself, all the connections have been tested and work fine. I am using a 16Mhz crystal.
The chip is fresh from Mouser, never been used. Would I need to adjust the fuses? If so, how?

-carjo24

(Whoops, missed the note about the crystal freq in your post above. )

Fuse adjustments would be needed later if you wanted to switch from using the 8MHz internal oscillator to the crystal, but if you can't get avrdude to talk to the chip you can't change the fuses anyway.
Using the Arduino IDE to program a chip can change the fuse settings and cause it to act similar to the manner you described, but you have a crystal attached so it should be a moot point.

Not sure what else to try at this point. Since all the connections appear to be good and the board is getting power I'd start suspecting component failure or a hidden solder short somewhere.

I am using an ATmega328p-AU. I am powering it with a 2 AA battery pack and using the 6 pin ISP header. The target power option is disconnected.

The target board I'm using is just a breakout board that's plugged into a breadboard. I made the breakout board myself, all the connections have been tested and work fine. I am using a 16Mhz crystal.

That will not work. Per the datasheet, only operation up to 10 MHz is permitted when running from 3 V DC. For 16 MHz, you'll need to provide 4.5 - 5 V DC.

I tried it with 5V and still get the same result. At this point I'm going to assume that the chip is bad. I'll buy a new chip and try again. Hopefully it works

It's rare indeed to actually get a bad chip, or damage it beyond repair. On the other hand, it's relatively easy to set the fuses such that it's expecting the wrong clock source, or something of a similar nature.