Torque is moment of inertia times angular acceleration. Theoretically, if you could provide a constant torque and there were no losses, you could accelerate to an infinite speed even with a small torque.
In reality, though, losses exist. Frictional losses are based solely on load weight, like static and dynamic friction in a bearing. There's also viscous damping, which is proportional to speed.
Given "enough time," your speed will equalize because the damping losses will exactly equal the applied torque. If you know how much torque you're applying and what the steady state speed is, then you can calculate the losses.
If you know what the damping constant is then you can calculate the ultimate speed for a given torque.
If you don't have either then there's nothing really you can do, though. You can try to get numbers from other systems but little differences in fiction could make the difference between 10,000 and 30,000 rpm, for example.