All down to the current... If you have a 2 amp wire connected to an AAA cell.... nothing (except the battery would go flat quickly). If the battery had 200 amps available, ie.. large car battery, the wire would just melt and burn.
If your wire was a 40 amp wire some damage may be experienced as there would be enough current to heat the acid in the battery....Remember.. A drop test (standard car battery test) has a 40amp shunt wire with a amp meter, to test for current loss.... To keep the wire from burning away you need to encapsulate in glass ie... a bulb.