How fast could a propeller driven piston powered plane theoretically go ?

[Jukka]

Yes shrouded prop could give the right answer (at low altitude).

Also the feather weight solar powered aerodyne by Paul McCready went 7 x faster than on sea level at 100 000 feet altitude.

[Exhaustgases]

Proof of concept. Prop will work at mach tip speed, and fly at mach speed. Look up the Thunderscreech.
And it flew faster than what most sites post.

[Jukka]

Proof of concept. Prop will work at mach tip speed, and fly at mach speed. Look up the Thunderscreech.
And it flew faster than what most sites post.

Yes someone could make a proof of concept. It is basically a thrust to weight ratio question. 4500 hps and 2000 lbs should be adequate. Little bit wing sweep and good Meredith effect and everything in balance and you are good to go.

[Exhaustgases]

I wish there was the finances so us like minded folks could work together and make this happen.