What is the Maximum Speed that can be acheived Because of Acceleration Due to Gravity? Planets and other bodies in the universe have different ranges of gravitational fields. What is the maximum speed that can be achieved because of acceleration due to gravitational field. 

In other words which is the strongest and largest gravitational field discovered in the universe till date. What is the maximum speed An object accelerating in these fields can gain?
It all depends on your frame of reference, but the biggest speeds caused by gravity will involve black holes or perhaps neutron stars.

From the point of view of a far external observer, an infalling object (from infinity) reaches a maximum inward speed of $0.385c$ (where $c$ is the speed of light) towards the black hole at 3 times the Schwarzschild radius. Thereafter, the object will appear to slow down and tend towards being stationary at the event horizon.

From the point of view of a "shell observer", that is somehow held stationary in the gravitational field of the black hole, the infalling object would have a speed that tended towards $c$ as the shell observer tended towards the event horizon at the Schwarzschild radius. It is meaningless to try to extend this inside the event horizon since (stationary) shell observers cannot exist inside the event horizon.