What is the maximum speed that can be achieved because of acceleration due to gravity?
The speed of light. But there's a catch.
In other words which is the strongest and largest gravitational field discovered in the universe till date.
That of a black hole. Sagittarius A* is thought to be the location of a supermassive black hole.
What is the maximum speed an object accelerating in these fields can gain?
The speed of light. See this list. The escape velocity for a black hole is the speed of light, and you can flip this around. If you drop an object from a great height, it's travelling at escape velocity when it reaches the gravitating body. But like I said, there's a catch. Take a look at this by Einstein:
See the second paragraph. The body falls down because the speed of light is spatially variable. If this continued unabated there would come a point where the body is falling faster than the speed of light at that location. But since matter can't faster than this the "coordinate" speed of light, the maximum speed it can fall is at the speed of light at that location. Which is circa half the speed of light at our location.