Correct answer: it depends.

But this is actually a fun thought experiment so lets make some assumptions. Lets try and launch a ~one foot diameter sphere that weighs about 50 kg to a height of ~330 km.

I just so happen to be working on a numerical simulation program for rocket flight to test a bunch of possible launch configurations for theoretical rockets. Using this will be much easier trying to solve an equation for the motion of a body through an atmosphere with changing density.

I am still using a very rudimentary aerodynamics engine until I can figure out something better. But it should at least give us order-of-magnitude estimates.

If I plug in a sea level launch and keep increasing the initial vertical velocity it has to go 22,800 m/s to get to about 324km. That is 51,000 miles per hour, or Mach 67. That seems pretty unrealistic. Granted it will only be in the atmosphere for a few seconds at that speed. It will feel a max of about 8000 g's of acceleration due to drag.

However if we launch at "from a hight mountain" at 4000 m (13,000 feet) that number comes down to about 9500 m/s (21,000 mph) or about Mach 30 (at that altitude) and a peak accel of about 1400 g's.

If we take a best possible situation approach and launch from the top of Mt. Everest (8848 m) you can get away with something like 4600 m/s (10,000 mph) or Mach 15! Since there isn't much air left up there the peak accel is only about 210 g's.

Obviously building a device to shoot a projectile 10,000 mph straight up at the top of Mt. Everest isn't any more practical than launching a rocket, but that's what thought experiments are for.