Is there any reason to believe that any measure of loudness (e.g. sound pressure) might have an upper boundary, similar to upper limit (c) of the speed of mass?
Yes - there is a sound pressure limit for undistorted sound. Over that limit we have a shock wave. It depends on the environmental pressure, but there is a theoretical limit to loudness which you can find here.
The limit is basically equal to the pressure.
Theoretical limit for undistorted sound at 1 atmosphere environmental pressure 101,325 Pa ~194.094 dB
The lower limit of audibility is therefore defined as 0 dB, but the upper limit is not as clearly defined. While 1 atm (191 dB) is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere, larger sound waves can be present in other atmospheres, or on Earth in the form of shock waves.
Well, the short answer is: there is, when the hydrodynamic approximation (that fluid is composed of small "fluid particles" i which real particles move in the reference frame of the "fluid particle" like in stationary fluid) breaks.
The upper bound can be approximated with wave amplitude equal to ambient pressure, so that the pressure is going down to 0 in wave minimas (this plus minus corresponds to cavitation); yet this corresponds to loudness of $\infty$dB.
Let's talk about the sound waves in the air first. Physically they are longitudinal waves of pressure. The bunch of air in one place will get compressed (in comparison with equilibrium state) and after that will expand, compressing the adjacent air and so on the wave propagates. These (single frequency) waves are essentially described by three numbers: the speed of propagation (called speed of sound; it depends on the type of the material and the temperature), the frequency of oscillation (determining the pitch), and the amplitude of oscillation.
It is the amplitude of oscillations which determines the loudness. So you are essentially asking whether there is any limit on amplitude of compression. Well, of course. At high enough pressures, the air would freeze even at normal temperatures. So this is the limit for air. Similar thing would happen with liquids: at certain pressures they would condense into solids. You could continue with phases of matter in this way and applying higher and higher pressures you would eventually end up with a black hole. That would be an ultimate limit.
But of course, in reality the limit is set by our engineering capabilities and I doubt it's possible to create sound waves that would be able to freeze air.
The only practical limit to sound-pressure might be when the medium were to be compressed into a black hole. Although I do not know about the sound-propagating features of black holes. Long before that the medium would come apart, f.e. into plasma.
After all, you compare two things: an upper limit of a speed with a power. How much power can you put into a particle to speed it up with c in mind?
PS: I'd rather not guess what kind of sound-generator would be necessary. One might ask the band "Disaster Area" (https://secure.wikimedia.org/wikipedia/en/wiki/List_of_minor_The_Hitchhiker%27s_Guide_to_the_Galaxy_characters#Hotblack_Desiato)