Note: All of this needs way more details than given. starting from model and modification all the way to cartridge used and level of backlight. So this is more a rule of thumb issue, which may need adaption to your use case.
Technically I guess it can draw as much as the source delivers - we're talking low current/power anyway. The basic GBA is powered by two (L)R6 (AA) batteries. At an internal resistance of 0.5 Ohm a current of 3A could be drawn by shortening the battery. Of course it'll be way less due additional resistance within the device. I guess the maximum ability will range around 200-300 mA (*1).
A basic 0.25 mm (10 mil) trace can carry at least 1A without being stressed, so already past whatever batter
It may be way more practical to look at two corner cases:
- A standard GBA draws about 60-80 mA during normal operation with a basic cartridge.
- At 200 mA the even Alkaline LR6 will be drained in less than 8 hours (*2)
This means that up to 100 mA would be a good assumption of what might be usable. Flash based cartridges with up to 80 mA consumption during operation support this.
As said, this will differ widely depending on the device and battery solution. The devices mentioned can easy span a 1:10 ratio for this topic, meaning that above 100 mA could as well be 1000 mA with the right device and battery pack/power adaptor.
If this is about construction some new device, your goal should be to draw as little as possible, no matter hoch much could be drawn. Not just to not damage the GBA or any attached device, but way more to be able to play more than a short time. This is not only die them being 'emptied', but more so due dropping output. With a higher demand the batteries will soon reach a level where they may still be way from being empty, but already insufficient from providing enough power to generate the current demanded:
(Taken from Wikipedia)
So once again, it all depends on the battery you're using. Best to assume the worst...
There is not really a 'too much' Szenario, as the GBA will not deliver 'too much'. There is a too less szenario and its not about the GBA, but your device.
P.S.: Comments suggest that this is about some add on game module intended for as high as possible power, so why rely on the GBA power at all? Wouldn't it result in way more freedom if the cartridge brings it's own power source? That way you're able to optimize (maximize) everything from voltage to current for the need of this module but are free of all power constrains by the GBA and free of taking care for various configurations and models. They should always be able to power themself, so no need to consider that.
There are essentially three ways to do so:
Add a (USB?) connector for an external power supply. Beside this being needed anyway for a SoC, the board can detect by default run in a low power more (like 200 mA max) and full power when USB is connected (500 mA at 5V (aka 600+ at 3.3) or more).
Find some LiPo pack that fits the cartridge in addition to a slim line SMD PCB. ~1000 mAh at 3.7V seems possible.
If power need is intermittent, a reasonable large gold cap can be used to bridge these sections of high power demand.
In fact, #2 and #3 could be combined (if the use case permits) to optimize peak performance.
Possibilities are endless :))
*1 - It's sufficient to just look at the current as the difference between 2.8..3V of the batteries and 3.3 after conversion is neglectable (<20%)
*2 - Quite visible with improved display mods