The short answer is about 150mA.
From the limited testing that I've done, the component you pointed out appears to be a positive temperature coefficient thermistor. Here's how Wikipedia describes their use as current limiters:
When first connected to a voltage source, a large current
corresponding to the low, cold, resistance flows, but as the
thermistor self-heats, the current is reduced until a limiting current
(and corresponding peak device temperature) is reached. The
current-limiting effect can replace fuses.
I only have a basic multimeter and a few fixed voltages handy at the moment, so I'm not able to derive the precise characteristics of this device. I was able to measure the following:
- Open circuit resistance is 4 ohms
- Short circuit current at 12V is ~50ma.
- Short circuit current at 10V is ~50ma.
- Short circuit current at 8.3V is ~60ma.
- Short circuit current at 5V is ~100ma.
- Short circuit current at 3.3V is ~120ma.
The tolerances on these parts are pretty loose, but for mine the maximum it can supply consistently in series with a motor is about 150mA. If I keep current above this for long, the thermistor begins to heat and total current decreases due to the limiting.
It might seem odd that the max constant current I was able to sustain is greater than the short circuit current, but this is due to the fact that this component is effectively limiting the amount of power it dissipates, and not the amount of current it passes. Power is the product of current and voltage, so because some voltage is dropped by the load, it is able to handle a current higher when loaded than it is able to when shorted.
As a point of reference if milliamps aren't very familiar to you, in my testing I was able to cause the clutch gear (60c01) to slip indefinitely on 43362c01 with the thermistor in series.
This drew about 100mA for me. So, you can drive most of the lighter 9v parts just fine. The current limiting will kick in at about half the stalled current for most of the motors. Larger PF motors, and especially the RC buggy motor, will not be able to reach their full potential with the overload protection in place on this battery box.