Ha I like your enthusiasm Peterthinks! but unfortunately in the US the legal limit for flying a multirotor is 400 ft. They can certainly fly higher than that, but I don't think Flitetest would like to show themselves breaking the rules. David did take an RC airplane to "space" for which he got special FAA permission to do. Its a really good video if you want to check it out. Flitetest could try to get that kind of permission again for a multirotor, but with today's sensationalism surrounding "drones" I bet that permission would be hard to get. If you really want to know the answer though you could always rely on good old science and calculate the altitude to which the thrust provided by the multirotor (which decreases as the air gets thinner) becomes less than the weight of the multirotor. And to answer the question about the rocket, if you're talking about commercial model rockets like Estes, a multirotor could certainly go higher. If you're talking about large scale rockets used for space missions, then there is no competition because the thrust of a rocket is dependent on the engines and fuel burnt and not the air density (maybe it is a little, I'm not a rocket scientist) whereas a multirotor's thrust is based entirely on the propellers churning through the air. So as the atmosphere gets thinner a multirotor will struggle to keep itself hovering while a rocket will keep on going (assuming it doesn't run out of fuel). Hope this helps!
They've already done an 'FPV to space and back' thing, but that was a few years ago before all the hype and controversy regarding the hobby. Unfortunately, with all the current issues I don't think it would be possible or sensible right now.
Never thought about the legal U.S. height limit. Yes I know the air gets thin and that would set a physical limit. Maybe there is a place where they could do this. Perhaps offshore in international waters? No rules out there. Lets bring this out to the pirates!
Unless you balloon it up there isn't going to be any way to get it there.
First you have the battery limits. Central Ohio is about 365 meters above sea level and space is approximately 100km above sea level. The multi-rotor would have to travel 99,635 meters. At an ascension rate of lets say 5 meters per second you would theoretically reach space in about 5.5 hours. No turbulence, straight up, and speed remained constant. Even the most efficient multi-rotors can only cruise for around an hour. Ascending takes a lot more power.
Next would be the fact that the props would become less and less effective as the multi-rotor climbs. There is less and less air that the multi-rotor can physically move to propel it upward. Very likely the multi-rotor would reach a point and even at full throttle not be able to climb anymore.
Multirotors can get up above the clouds but the trip up there will take most of it's battery. Here is an example of it and what can go wrong.
Physics limitations aside, so? They could leave the US. Take a trip somewhere beautiful that doesn't have laws for this at the moment, do this experiment and use the opportunity to do some epic location filming.
Realistically for example, go up a mountain. do the height test from the peak and while you're there do flybys and races down the mountain again.
MAAC rules in Canada also forbid altitude above 400ft and attaching rocket engines to your rc. If you want to get to space your going to have to join a rocket club. Multirotor aircraft is a poor choice for altitude anyway if you could get permission to fly high you would have better results with something like an Avro Arrow or U2 with ducted fan engines or turbines