If g (acceleration due to gravity) was double on Earth, how tall would our atmosphere be?
Assuming no change in the total atmospheric mass or temperature, the Atmosphere would be half as thick.
Actually, most atmospheric scientists give no thought to how high the atmosphere might be. The assumption is that it grades imperceptibly into outer space, with no fixed interface. It does not appear to be a question whose answer would have any practical value.
Or, we might treat it like the old riddle, "How long is a piece of string?"
The correct answer is, "Twice the distance from the middle to one end!"
The "scale height" is useful, though, along with other concepts related to the OP's question such as the top of the tropopause.
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