Is there a constant power supply?

A voltage source is a device that produces constant voltage, but allows amperage and power to vary, in accordance with the laws of electricity. A current source is a device that produces constant current, allowing voltage and power to vary.

Is there a constant power source – i.e., a device whose output power never varies? No matter the nature of what it is connected to, its voltage and amperage would be adjusted to provide constant power. (Its behavior would be undefined for both open and broken circuits.)

Answer

Yes, it's very easy to construct a constant-power supply.

Take, for example, an ordinary switchmode boost converter.

^{simulate this circuit – Schematic created using CircuitLab}

Let's assume it is operating in discontinuous mode and does not have synchronous rectification (i.e., just a diode). If the switch is operated with a fixed duty cycle (i.e., no feedback), it puts a fixed amount of energy into the inductor each time it is closed. The amount of energy depends only on the input voltage, the inductance and the on-time. That energy gets dumped into the load when the switch opens.

Constant energy per cycle × constant number of cycles per second = constant energy per second = constant power.

Regardless of the resistance of the load, the voltage and current levels will adjust themselves to match that value of power.

In terms of practical limits, if the output of this supply is shorted, then the current will be limited by the resistance of the internal components (the inductor and diode). If the output is left open, the voltage will be limited by the distributed capacitance of the components — the inductor will "ring" with some high voltage at the self-resonant frequency.