If I looped + and - terminals of a single AA 1.5 V, 2 Ah battery with a broad wire
You'll never get such a high current from an alkaline/ni-mh battery because the internal resistance will be too high (0.1 - 1ohm, possible even more). You can usually short such a battery for a brief period and it should not cause damage, although I wouldn't advise it in any case since the wire will get very hot and you may end up burning yourself.

Battery resistance may vary from manufacturer to manufacturer, but a rough guide can be found here: 
http://www.learningaboutelectronics.com/Articles/Battery-internal-resistance