There are different definitions depending on who you ask. I think NASA had one definition that was about 50 miles up – this means that their X15 rocket plane pilots could be recognised as astronauts. The common definition these days is 100 km I think, but it is kind of arbitrary (there is still a bit of atmosphere even at the 200 km altitude of the international space station).
A weather ballon might get to 40km or so, not quite as high as the X15, but still very high.
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