How high can propeller airplanes fly?
It is a matter of power source (and engine aspiration in case of air breathing engines), wing loading and aerodynamic efficiency. With current technology, the limit is around 100.000 ft (30 km), as proven by Pathfinder and especially Helios. I doubt that much more is possible with really useable aircraft.
Aerodynamics first: The altitude factor of $c_l \cdot Ma^2$ tells you how much lift can be produced at a given flight Mach number, and the wing loading then gives you the minimum density for sustained flight. 0.4 is a good value for subsonic $c_l \cdot Ma^2$, and 30 kg/m$^2$ is a feasible wing loading for flight at 30 km. See this answer for more detail.
If the power source needs ambient air (piston engine), the plane needs triple-stage compressors or turbochargers, which have been tested up to 20 km altitude and should be good for maybe 24 km. They are finicky devices; Boeing Condor rarely flew at its maximum power because the stages of the turbochargers would oscillate in an alternating sequence of surges. One stage would race ahead, causing the other to surge, which made the first surge and free up the other to race ahead, and so on.
Above approximately 24 km, solar-electric propulsion looks like the best option currently. In all cases you can only fly subsonic, so the minimum practical wing loading will limit the maximum altitude. Aircraft like Helios are very delicate already, so they can only be launched in calm weather and are at risk of being blown away by high altitude winds. Payload is minimal, and depending what the aircraft is supposed to do besides flying high will give you a limit on the maximum altitude between 24 and 30 km.
Going to orbit in a propeller driven device is completely illusory. There is not enough matter to push against at higher altitudes, and the theoretical propeller diameter would be measured in Kilometers (or miles, if you prefer that unit). The structural mass would be prohibitive. Also, propeller thrust is inversely proportional to flight speed, and there is no way to accelerate with a propeller to escape velocity. The thrust would be just a rounding error away from zero at 7.9 km/s.
This speed is required to escape earth's gravity by flying fast enough around it so that the centripetal force equals the aircraft's weight and is called orbital velocity. The higher the orbit becomes, the more energy is required to reach it. In order to gain enough energy, a propeller aircraft would accelerate in the atmosphere to a speed quite a bit higher than the desired orbital velocity and then convert that kinetic energy to potential energy to lift its trajectory above at least 100 km, the internationally recognized altitude where spaceflight starts. Note that this phase of the flight requires inverted flight if the acceleration takes some time. The maximum flight Mach number would need to be maybe 12 or even 15 so this maneuver is possible at all.
In short: Going to orbit with a propeller? Forget it!