What Determines The Amount Of "Current" In An Amp?

Some speakers which dip to very low impedances at certain frequencies don't require lots of current, but rather an amp that is stable into a highly-reactive load that presents an almost dead-short operating condition, with, as Viridian mentioned, wild phase angles, a combination that make some amps oscillate in a state of instability. One such speaker is the original Quad, which was sometimes faulted for the sins of the amp driving it.

If you ain't got enough current, I can sell some Amperes for you (i can pour'em to the medicine bottle); trade-ins are OK as well.

05-11-15: Mitch4t
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Thanks Viridian.

In the very big amps of old, they boasted large power supplies. Is it necessary for Class D amps to have a large power supply to deliver continuous high current into low ohms?

In general, yes, SMPS power supplies are capable of supplying large amounts of current into a low impedance. Today one can see several class-D amps capable of 20-30A into a low impedance such as 2 Ohms. Bill_k already cited the Merrill Veritas. There are a few others too - names are eluding me right now....
In a SMPS the amount of power delivered depends on the size of the power inductor & the size of the output power switches (that switch with PWM control, most often). Ultimately, in SMPS power supplies too the amount of output current is limited by the size of the AC power transformer because SMPS too have to take as input the AC from the wall outlet, rectify it to DC & regulate that for the class-D power amp. Whatever current the class-D power amp needs comes from the SMPS power supply & in turn the SMPS power supply takes its current from the wall outlet i.e. AC power transformer.
So, high current is possible thru a SMPS-based class-D power amp - just have to size the AC power transformer correctly (given that the SMPS design is correct for high current).
Many class-D amps are stated to be stable down to 2 Ohms but I see very few capable of stability into 1 Ohm. Part of the reason could be that very few 1 Ohm speakers exist today.