I understand that in nature wind would never get high enough, but I am just curious as to whether physics would allow this to occur or not.
Answer
I think that some confusion is here. Wind speed is determined by differences in the air pression between two points. The max. limits that we observe must set the max wind speed. Wind is the movement of a mass of air. The Sound speed determines the sound barrier. Sound speed is determined by considerations about density, temperature... and is a property about the relative speed of the wavefront of a sound event (a perturbation of te medium - air, that propagates) in relation to te the air (considered at rest). The speed of sound in relation to the ground is the vector addition/subtraction of the sound speed (+-340m/s) with the speed of wind in relation to the ground. The max wind speed is observed in the jet streams in altitude and in the tornados and hurricanes at surface. Somewhere we can find the max speed of the exhausted air in the jets and determine if air moves trhu the air with a speed superior to the speed barrier. Nothing prevent this from happening with enough thrust. I've no time to search now, sorry.
EDIT add:
There is is need to search. The exhausted mixture of jet engines, not beeing wind but mostly a mass of air, can travel thru the atmosphere at speeds much superior of sound barrier. If it is was no so the jet planes could not cross the sound barrier.
So, inspite of the confusion in the question, I the short answer is a Yes (a mass of air can travel faster than max sound speed).
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