I want to create a system inside of the drivetrain. So basically 2 of the wheels are powered by green motors which are geared to a 3:5 gear ratio which makes an RPM of 333. So from my understanding of both the external and internal gearing of 18:1, is 3596.4 RPM. However, if since these wheels are not being back driven, this would not be a problem. However, I want to add another blue motor to this design which powers another wheel. I want this blue motor to have any RPM higher than 333. So how this would work is, when the bot reaches its max speed with the green motors, the green motors would coast and the blue motor would kick in. This makes it similar to a 2 speed gearbox in a way. So, here the problem comes in. Lets say for example, my blue motor has a 1:1 gear ratio producing 600 RPM. Whenever the blue motor is running, the green motor is being back driven. So if we go back and do the math, at 600 RPM, the green motor is being back driven at 6480 RPM. So when making the design, I need to account for back driven motors like these in order to create the most effective but also physically possible drivetrain. But, I have found no information on the internal back driven RPM limit of either the green or blue motor. I am wondering if anyone has information on this limit.
TLDR: Don’t mix motors with different gear ratios in the single drivetrain! 
There are many reasons why mixing motors like that is a bad idea. Just a few of those:
- Since vex motors have permanent magnet stators they act like generators and, if you backdrive them at twice the nominal speed, they will generate voltage twice the nominal supply voltage, which could damage electrical components up the circuit.
- When green motors are backdriven they will not merely coast but will cause a lot of mechanical resistance, consuming usable power from the blue motor.
- Since VRC has constrained number of available motors, then for a competition robot you want to maximize usage of every motor at every point of time. So having any of the motors idling is suboptimal.
Your best bet would be to use all three motors identically geared around 400-450 RPM simultaneously, since they don’t generate max torque and output power at max speed anyway:
Also, from the thermal load standpoint (to avoid motors overheating) you want to run them at less than 65% max speed most of the time with short bursts up to 100% of nominal speed and power occasionally.
However, if you insist on having an option of either very high torque or very high speed with the minimum number of motors, you could try building pneumatically shifted transmission or even some sort automatic transmission but be warned that it could get overcomplicated really fast.