The max speed of acceleration would be from Zero directly to the speed of light, nothing can travel faster than the speed of light, the energy required to do that feat for anything with mass would be infinite, lets say you had a mass of 1 kilogram as our "Test Mass", the max acceleration would be from 0 to C which is an acceleration of 299,792,458 m/s^2, the fastest you can accelerate because at the point of the speed of light your acceleration would drop to zero. This is because the Force and Energy required to accelerate yourself past the speed of light would be infinite due to Relative mass under GR, which has been proven to be correct.
Solve this equation for traveling at the speed of light, you will find that no matter your original mass your mass at the speed of light is indeed infinite or undefined, the measurements even go insane as you approach the speed of light to almost infinite numbers.
If you remember, that Force is defined as Mass * Acceleration, then to accelerate a infinite mass it would take a infinite force to accelerate it at all. Secondly, you may say well, I can accelerate to the speed of light in as little time as I want then, well no there is limit in our universe to the smallest amount of time which is Plank Time, which is the smallest unit of time possible in the universe made from natural units. This being about 5.39106*10^-44 seconds a very short amount of time.
So, you could in theory accelerate to the speed of light in one plank time which would require a near infinite amount of Force, but in theory the max acceleration of a object is 1.03 * 10 ^ 95 m/s^2 if you were to accelerate directly from Zero to the speed of light in one plank time or 2.06*10^95 m/tp^2 if you were going the speed of light in the direction then reversed to go the speed of light in the 180 degree direction, Now you would think this is correct, but still not quite there you have failed to realize that converting to a different set of units makes this not balanced because now we are in plank units, So you have to count in Plank lengths, which Lp/tp = C which makes your max acceleration still 299,792,458 Lp/tp^2