generator coils: max power given wire gauge
Generators, transformers, motors, all have the same relationship for the maximum power of a winding.

The amount of power you can safely put through a winding depends on the amount of copper you use, measured either as mass or volume. It doesn't matter whether it's a few turns of thick wire, or many urns of thin wire, the current carrying capacity varies in inverse proportion to its working voltage, to yield constant power.

You can do a simple thought experiment with a winding space that has two identical coils of N turns wound on it. Put them in series for 2N turns of thin wire, put them in parallel for N turns of wire with twice the area. The same volts/amps on each individual coil adds up to the same total power handling for the same power dissipation within the coil, while the voltage and current at the terminals change by a factor of 2 depending on the connection.

While the voltage rating is quite easy to measure or compute, the current carrying capacity is more of a 'ratings' exercise. You have to do two things, (a) keep the windings cool enough and (b) keep voltage drops due to \$V=IR\$ within your specification. 

For a short duty cycle, you can run higher currents than if you want to run continuously. As a rule of thumb, small to medium power transformers and motors tend to run about 3 A/mm² current density, but that can change substantially with different cooling arrangements or different interpretations of what small and medium mean. Fortunately it's quite easy to measure winding temperature when assessing how much current you can use over how much time. The resistance of copper increases by about 10% for every 25°C rise in temperature. Measure the winding resistance at room temperature, and then again after a period of running. I tend not to like exceeding 50°C rise, but then I'm a cautious soul.