Once an object reaches terminal velocity in free fall, is there a point in time when the object actually starts slowing down due to resistance, or does it stay constant?
As @Ruben explains, if air resistance and gravity are constant, the terminal speed will stay constant too.
However, while you are falling down, the air density increases, reducing your speed. A very good example of this is Felix Baumgartner's fall from almost 39,000 m. His highest speed was about 1350 km/h, but by the time he opened his parachute, he was only doing about 200 km/h - the normal terminal velocity. This site gives a graph of his speed during the fall. As you can see, it increased rapidly to about 350 m/s, and then decreased to 50 m/s. This deceleration is due to increased air resistance.
During the fall, the rising gravity also increases your speed, but this effect is much smaller than that caused by air resistance.
Terminal velocity is the speed at which the drag force equals $mg$; hence, the object will remain at that speed.
That said, there will be slight fluctuations because of the increasing strength of the gravitational field as well as the increasing density of the atmosphere.