Hm, I'm curious here. Why would one not want the tip of the prop to breach the
speed of sound? Would it have something to do with the rest of the prop not
spinning as fast? What would happen if it did?
Supersonic prop tip speeds do occur. T-6 Texan props are one example and if you've ever been to Reno and heard the T-6 class race, you know that they have a very loud prop noise. Some Cessna 180 owners have the pitch on their props so high (in order to generate a lot of static thrust that shortens takeoff distance) that the prop tips will go supersonic resulting in a lot of noise on takeoff. The Russian "Bear" bomber reportedly has very loud prop noise as a result of its props going supersonic.
Again, generally speaking, supersonic prop tips can create three conditions: (1) Extremely loud noise. (2) Flow over the propellor blade separates, greatly reducing prop efficiency and creating drag and (3) introduction of vibration/stresses on the propeller blades that can damage the propellor. Generally speaking, the aircraft won't gain any advantages from the increased prop speed/diameter if the blade tips are supersonic, and therefore there's no reason to design a prop that will regularly have supersonic tip speeds. There are exceptions, as I mentioned above.
Not the best example, but watch this 180 video and you can hear the prop tips going supersonic during the takeoff and climb out.
http://www.youtube.com/watch?v=_BI0c_0uupQ[EDIT] Here's a video of the T-6 class at Reno. A better example of supersonic tip speed noise.
http://www.youtube.com/watch?v=nen94IP8mNg