Finding maximum current experimentally
You need to note the difference between "can provide" and "is designed to provide". I'll call the latter "should" provide.

Many device can provide much more current than they should provide. Wll engineered devices will be equipped with protection to prevent attempts at overload causing damage. A typical example of this is a fuse - which "blows" when a circuit provides several times the current it should.

Whether something is damaged by current overload depends very much on the device and on the designers ability and/or the application. As a general guide, determining current by overloading is not likely to be wise and will often cause damage - often catastrophic but sometimes paertial. Partial damage may not be immediately obvious and may be cumulative.

Well designed power supplies will not be damaged by current overload. Maximum current is usually into a "short circuit" although some supplies use "current foldback" to reduce fault current until the overload is removed. More usual is for the supply to drop it's output to a low value. Or a fuse or circuit breaker may be fitted.

The maximum current that can be drawn from AC mains is set by design - usually breakers or fusing at each level. Without these a domestic outlet could provide 100+ amps and lots of smoke.

Small batteries may suffer a degree of damage if temporarily shorted. High energy batteries such as NimH and LiIon may melt or catch fire or come close to exploding - especially LiIon.

Shorting a car battery MAY be the last thing you ever do.