As the battery begins source current, it heats up and the chemicals inside begin to polarize, causing the internal resistance to increase significantly reducing it from the 15A theoretical to the realistic DC output of a few Amps max.
Chemistries like Lithium Ion can pump out far more amps for longer, because the chemistry does not break down so easily as those in Alkaline and Nikel Metal Hydride battery chemistries.
Another issue is the voltage of the AA battery while under such a huge load will sag significantly. Total rated capacity also takes a massive hit for certain chemistries, meaning sustained loads that large will drain the battery 5+ times faster than expected.
Again, Lithium Ion chemistries are more resilient to voltage sag and capacity loss under load. Unfortunately though, Lithium Ion batteries tend to explode into fiery awesomeness so care must be taken - with great power [density] comes great responsibility.