So far, no one has mentioned the hard limit of propeller tip speeds. You have to take the static tip speed of the prop at redline RPM (or whatever RPM you're using at the max speed) and square it. Then you take the forward speed of the airplane and square that. Add the two results and find the square root, and you have the propeller's tip speed at the max forward speed. A squared plus B squared = C squared.
Now, propellers do not do well anywhere near the speed of sound. Remember that the airflow over an airfoil accelerates over the top of it, and that happens to a propeller blade, too, so nearing the speed of sound a shock wave forms at the prop tip as that accelerated airlow gets to sonic speed, and drag increases hugely and noise gets awesome. Just listen to a Cessna 185 with a seaplane prop on it during takeoff. That constant-speed prop is at redline (2850 RPM) and the tips of the 86-inch prop are doing 729 MPH plus the small factor of the forward speed of 80 MPH added to give a tip speed of 733 MPH. Speed of sound at sea level and standard temperature is 760 MPH. Some 185 pilots find that they can dial the prop back and leave the throttle full and actually get better climb and less noise, since the drag on the prop has decreased and more HP is going into performance instead of noise.
Now, some airplane doing 500 MPH, like an old warbird all hopped up and spinning a 12-foot prop, has to have a low RPM and huge pitch. 1200 RPM, maybe. And the picture of the Bearcat with that massive prop shows what such a prop has to look like to absorb all that horsepower at such low RPMs.
Now you know why jet engines rule for speed. And why helicopters are relatively slow.