Heres my attempt:
First, Gravity is an outside force. Therefore no force is applied to it, and not being able to accelerate any further. So no, an object acted upon by no outside force will not double in speed until it reaches the speed of light.
Second, if gravity is not an outside force, and if that object has no resistance, the object will continue to accelerate at a decreasing amount until reaching near speed of light, at which point mass of that object has increased dramatically, will require an infinite amount of force to accelerate any further. At that point acceleration will become static, and cannot reach light speed.
Thats the real reason we can not go the speed of light, at least how I comprehend it.