Q:

Is an infinite Doppler effect plausible? 

When I was young (which I still am) I was amazed by the sound jet fighters make when they break the sound barrier - a sound similar to an explosion which caused my school to shake. 
Later I learned on school that this was the cause of the Doppler Effect.  Because the jet fighter approaches/reaches the speed of sound, the sound 'stacks'(?),  like so:

First question.  When I look at a jet which is about to break the sound barrier (v = 340m/s)and produces a constant noise of 120dB. How do I calculate the sound of the 'explosion' when the acceleration of the jet is for example 10m/s/s?  I know doubling the sound would make 123dB, but I don't know what the multiplier is if you break the sound barrier. 
Secondly, as jets making sound barriers already cause windows to shake, what would happen if a jet fighter producing 120dB constantly could maintain the exact speed of sound (so withstanding the shock waves and pressure).  Is there a limit to the sound it can produce? 

A:

So this is a many-part question.  If you look at that diagram, in fact, you will see that the pressure peaks and troughs (the circles) do not "stack up": nowhere are they intersecting.  So in fact the only effect in this diagram is that the pressure for the circle of radius $r$ is actually decreasing like $1/r$ as the circle gets bigger.  The intensity, which is a vector density going like $\vec I = p~\vec v$ where $\vec v$ is the velocity of the air particles.  Usually $|\vec v| = p / z_0$ where $z_0$ is the so-called acoustic impedance.  So surprisingly, the "usual" account says that the energy delivered does not depend much on frequency! 
With that said, right at the boundary, the core assumptions that let us derive the wave equation tend to fall apart, as the effective wavelength of the sound becomes lower than the mean free path of the air molecules etc.  So it's not just the Doppler effect: in the image that you've got, the "stacking intensity" is only happening at the rightmost part of the circle in the middle, where the Doppler shift is so strong that the wavelength drops way below some fundamental distance $D$.
The intensity of the wave is not going to go infinite, either: it is being carried away, in the reference frame of the jet fighter, two-dimensionally in all of the transverse directions.  So it's going to hit a maximum, no matter what. 
The "multiplier" when you break the sound barrier is actually very dependent on the object that tries to break the sound barrier, with this paper quoting a quarter-power-law as a function of aircraft weight once you start to factor in how the wingspans need to change to keep the aircraft flying etc.