How much current can you pull safely off a standard AAA battery?
The maximum current depends very much on the chemistry of the battery.

The capacity of the three main (no Lithium) batteries is approximately:


Zinc-Carbon: 540mAh
Alkaline: ~1000mAh
NiMH: ~900mAh


The current limit and capacity of any specific battery can be found in its datasheet.  For instance, the Duracell MN2400 has the following nice graph:



So you can see from that, 500mA would start to drop off significantly in voltage after around 1 hours.  1A would last about 30 minutes, if that.  So 700mA would be somewhere between there.

Also, from the datasheet, you can find the impedance.  250mΩ in this case. At 1.5V full charge you should be able to draw up to (I=V/R) around 6A from it. It'll probably not like it, and get rather warm, or explode, but 500mA - 1A should be no problem.