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.
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.
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).
Wind is caused by differences in pressure. Wikipedia's discussion of atmospheric Wind is here.
However I suspect that what is requested is whether the velocity can ever exceed the speed of sound for air > Mach 1 ie supersonic wind. Well there is such a thing as a Supersonic Wind Tunnel which is generated by pressure difference ratios of 10 or so. This corresponds to the use of a nozzle with those pressure ratios between input and output streams. There are also videos of objects in these tunnels.
The hypothetical Hypercane only has winds of around 800kph, so even this extreme situation is well below the soundspeed.
I'm not so sure that some gas giants might not have local winds in excess of the sound speed, here is wikipedia on Neptune "Neptune has extremely dynamic weather systems, including the highest wind speeds in the solar system, thought to be powered by the flow of internal heat. Typical winds in the banded equatorial region can possess speeds of around 350 m/s, while storm systems can have winds reaching up to around 900 m/s, nearly the speed of sound in Neptune's atmosphere. Several large storm systems have been identified, including the Great Dark Spot, a cyclonic storm system the size of Eurasia, the Scooter, a white cloud group further south than the Great Dark Spot, and the Wizard's eye/Dark Spot 2, a southern cyclonic storm."
I suspect Gas Giants can have high local wind speed with respect to the local speed of sound, because wind systems may extend to great depth/pressures, where the sound speed is much greater, so that the ratio of soundspeed at depth, versus soundspeed high in the atmosphere can become quite large.
Wind speed is the speed of an air current. The laws of physics don't have a privileged rest frame, so it can't have a bounded speed.
Imagine you're in a supersonic jet travelling faster than the speed of sound. From your point of view, you are stationary and the air is moving. Hence, the air appears to be moving at supersonic speeds.
I would suggest giving the subject matter a platform, obviously atmospheric wind currents bound within its own air pressure constraints could never naturally reach critical mass. However applying this theory to a directed finite point, one could suggest that using frequency adaptation, within a weapons platform similar to haarp technology, a high frequency air pressure canon could induce wind speeds with enough force to cause a sonic boom, good luck holding the weapon.