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Quite a lot, if you cool it to near-absolute zero : )
In normal conditions, you're limited to the point when the wire will just heat and melt.
Ordinarily semiconductor devices will have a rating, and that rating includes the package.
I would say 5A, it depends on the length of the lead and the thermal characteristics of what its soldered to. i.e. if its very short and soldered into a big area of copper on a printed circuit board, the copper will dissipate a lot of heat ,so keeping the lead cool, but while copper melts at 800 deg C the semiconductor at the end of it is only good for 150 degC.
Really depends on the device. Most common is 1-mil gold wire
if you're in plastic IC packaging, and this is good for about 1
amp tops (per wire). Power devices will be using thicker (2-10
mil) or ribbon bonding.
The leadframe itself is not the limiting factor in all likelihood.
I would bet that you'll find a max continuous current rating
for the specific device, if you look. This is probably driven by
something other than leadframe current capacity, but this
rollup is what ought to be respected - weakest link and all