After leaving the wires plugged in for 20 minutes at 50A, everything was still pretty cold. Being an engineer, the next obvious step was more current. I stopped at 63 Amps because that's much more than my controller can provide.
Note that I'm pushing 63 Amps through a single wire... there are three in the motor and on average my 50A BMC controller can only pump 50A between the three TOTAL, so I'm pushing a LOT more current than I'll actually use.
Also note that there's only a 71.09mV voltage drop across the wire!!! This means at 63 Amps the wire only consumes 0.318 Watts. That's insanely low. At 50 Amps, the wire drops 58.69mV and therefore consumes only 0.172 Watts.
One of the funny things about working at NI is that we have literally millions and millions of dollars worth of test equipment, but the hardest thing to find is always the cheap stuff: wires, thermocouples, fuses, etc... After digging through shelves full of $15,000 scopes, $3,500 DMMs $35,000 RF Generators, Downconverters, DAQ boards, FPGA boards, microcontrollers, touch screens, 24 bit DSA ADCs, etc, I finally found a box marked 'thermocouples'. This was no ordinary box... upon opening, I was greeted with at least 1,000 thermocouples in all flavors and variants. After sifting for a minute I found a nice J type specimen and took it back to my desk.
I hooked it up to a DMM, verified it returned the correct temperature with a precision heat source, and stuck it into the axle. I left the 63 Amps flowing through the wires during my entire thermocouple expedition, and after waiting another 10 minutes I measured:
39.04 degrees C. That's 102 degrees F, for those that don't buy into the metric system. That's much much MUCH colder than with the old wires, which would have melted into a pile with this current for this amount of time. It was, however, much warmer than I would have expected for only dropping 0.318 Watts. I quickly realized that the nonsoldered power supply current leads were burningly hot due to the resistance across the junction between the leads and the 10 gauge wire. With a DMM, I measured the voltage drop across this connection to be 0.2 volts, which is 25 Watts. So the heat I measured in the 10 gauge wire was due to the copper convecting it away from the connector. This won't be an issue on the bike because I'll solder the junctions together.
Next up, I decided to test out the current handling ability of the coils. Again, in real-world use, this is a polyphase AC motor, but I couldn't find an AC precision power supply that could output more than 2A. So I used the DC power supply, which doesn't make any difference in the power consumed since I'm nowhere near the skin frequency of the wires. I was more cautious this time and started my testing at 22A for 20 minutes across one of the coils. Same as before, in the real world I'll be dropping 50A max across three sets of coils, so 22A in one of the three sets of wires is already more than the motor controller will ever provide.

After 20 minutes, nothing had warmed up, so I moved the thermocouple in between two coils (one of them being energized) and kicked it up a notch at 30, 40, and then 50 Amps. I left 50 amps pumping through one phase for 20 minutes:
You can see that we've only risen to 55.7 degrees C (132.7 degrees F). The motor was warm to the touch, but by no means hot. 50 Amps through one of the three sets of coils is more than seven times the energy that the motor is rated to handle continuously out of the box, yet we took it like a champ. Obviously, the motor isn't inside of its enclosure, but I didn't have a fan blowing on it or anything to cool it off. Still, I imagine you could continuously run 50A through this motor with no problems with this wiring modification. I definitely WON'T be doing this, but it's possible. If the motor doesn't heat up at all at 1000W continuous, I might try to see if it can handle 1500W, but I really don't like traveling at the speeds this would necessitate.
So in summary, upgrade your wires to enameled, 10 gauge, solid core and current will not be an issue.