What is the fastest possible acceleration? Context & Question
In physics, the fastest possible speed is accepted as the speed of light. No other mass has been recorded to travel as fast as the photon. What is the acceleration of the photon? How long does it take for it to get to its top speed, from emission from its source to its constant velocity through the medium of space-time? Does the photon also hold the record for fastest acceleration? Taking these questions into account, I would like to probe further and ask if instantaneous acceleration is indeed possible by the laws of physics? 
 A: Quantum gravity arguments indicate that the smallest amount of time is of order the Planck time which is approximately $10^{-43}$ seconds. The speed of light is approximately $10^8$ meters per second and so the fastest acceleration is of order $10^8/10^{-43}=10^{51}$ meters per second$^2$.
A: Your questions is tricky (and interesting) since it is located on the boundary of established theories.
Following the assumptions of relativity theory: The photon never exists in a state where it has a different speed than the (local) speed of light for all reference frames. It "is born" like that. Since the photon is measured at the same speed from every frame our usual concepts of acceleration and speed break down and it is doubtful if extrapolation from below-light-speed makes sense.
Following the assumptions of quantum electro dynamics (QED): The photon does not take a single path which allows us to determine a speed or an acceleration. It takes (at least from the perspective of the employed mathematical procedure which comes up with the correct probabilities) all possible paths from emitter to detector at the same time, we must sum up all the probability amplitudes along all these paths and we get a probability at which detector the photon pops up into existence. Statements about paths taken make no sense since we cannot measure them in principle.
Probably there is such a thing like maximal acceleration. Accelerating an object for a certain distance requires energy in a small location near that object. Too much energy means too much mass. Too much localized mass produces a black hole. So there probably is a limit. Still: We are debating a situation where we are widely outside of the range in which our existing theories had been tested. So a break down of our physical concepts is quite likely as well. 
A: Pretext prior to start. Using specific scientific language is crucial when we talk to each other to avoid misunderstanding. We are all scientists here of varying complexity so we should remain focused on furthering knowledge and education without prejudice or ego.
“In physics, the fastest possible speed is accepted as the speed of light.” ~ Fastest observed velocity to date is that of a photon. There is a potentially hinted at faster motion with the arrangement of spin. See spooky action at a distance quantum entanglement.
“No other mass has been recorded to travel as fast as the photon.” The photon itself (as far as we can determine) is massless. But be aware that the m in E=mc2 doesn’t have to be mass it can also be ‘moment’ as well as potentially other forms of motion. This then translates to the potential for other dimensions having their own ability to take the place of m in a quantum calculation. Note* the speed of a photon is different in different mediums and we can get photons currently down to walking pace and other sun atomic particles can then overtake it.
“What is the acceleration of the photon?” ~ The acceleration of a photon is the exact same as it’s velocity.
“How long does it take for it to get to its top speed, from emission from its source to its constant velocity through the medium of space-time?” ~ so to expand on the above context is required…..; 1) A photon is a packet (fancy word for group) of different electromagnetic waves. These waves are comprised of two alternating ‘fields’ known separately as the electric field and the magnetic field. A changing electric field generates a changing magnetic field and vice versa. Think of it as a game of leap frog as each field goes from 0-1-0. as the electric field collapses from 1-0 it induces the magnetic field in front of it in space and the magnetic field goes 0-1-0 and then as the magnetic field collapses from 1-0 it induces an electric field in front of it in space so on and on for ever(as far as we know). To be clear an electromagnetic wave and a photon are essentially the same thing. A wave is singular, a photon is a group of waves. Space (as far as we know) is not a medium.
2)”medium of space”~ This trail of thought comes from a classical physics theory centring on a thing called the ‘ether’ which comes from an early lack of understanding in an attempt to use experience as a reference. There is no medium in space…..(that we know of)
3)The changing from an accelerating electric field to an accelerating magnetic field happens at the heady 299,792,458 metres per second.
4) If a photons (lights) motion was a ‘uniform’ (velocity) motion then its measurable velocity would be different for different observers in different velocities relative to the direction of the photon and the observer.
But….
Changing from positive to negative and from electric to magnetic is a changing motion (aka acceleration). The photon is massless enabling instantaneous acceleration, this then gives an average motion (velocity) equal to its acceleration.
“Does the photon also hold the record for fastest acceleration?” ~ Soooo…. Fastest has several possible definitions. A. Velocity B. Acceleration C.  Velocity over a given distance D. Velocity through a specified medium (or lack of). A photon is the fastest motion we have observed through space(a frictionless medium).
When a photon goes through a medium it is absorbed by the electron of an atom it is then absorbed and then ejected only to be absorbed and ejected over and over again. Different wavelengths enter and exit at different angles based on the medium due to sympathetic wavelength. Meaning that certain wavelengths are more or less likely to be absorbed based on the mediums constituent elements and higher wavelengths of a field have more a more defined probability which means the area where they are likely to be found as a point like particle is smaller than that of a larger wave thereby reducing the overall ‘area’ that they can interact with an electron.
“Taking these questions into account, I would like to probe further and ask if instantaneous acceleration is indeed possible by the laws of physics?” ~ answered above 
A: If a photon can be said to change speed at a boundary between two optical materials (e.g., glass and air), then you might ask how quickly it changes speed at that boundary.  However, no physical boundary is infinitesimally thin, so at least on a sub-femtosecond timescale, the effective speed of a photon changes somewhat gradually while crossing a physical boundary.  But as the other answers have indicated, the "speed of a photon" is not precisely definable in a physically useful way because a photon is not really a point particle; it's a spread-out wavepacket.  If the speed can't be defined precisely, the acceleration can's be defined precisely.  
You didn't ask about acceleration of a point charge like an electron.  We can come a lot closer to defining velocity and acceleration for an electron, but ultimately run into the same limitations due to the fact that an electron, too, is really a wavepacket.  However in the case of an electron there is a further limitation: an accelerating charge radiates EM radiation at a rate proportional to the square of its acceleration  See Larmor Radiation.  So, the quicker you try to change its speed, the more it will cost you in expended energy.  It would take an infinite amount of power to provide infinite acceleration to even a massless point charge if there were such a thing.
The speed of light (which doesn't have quite the same meaning as the speed of a photon) is a property of the vacuum of space, "inhabited" by electromagnetic fields described by Maxwell's equations.  An electromagnetic wave (not quite the same thing as a photon) moves at a speed determined by the properties of the matter in space the wave is moving through, so "acceleration" of an EM wave (i.e., change of speed or direction) can always be traced directly to a change in material properties along the spread-out "path" of the wave.
