[reorganize]
I'm going to "fix up" my original reply some, to make it simpler and more clear by rearranging it a bit.
Objects with escape velocities greater than the speed of light do exist - they are called black holes.
There is a sense in which when one falls into a black hole that one moves "at the speed of light" at the event horizon. Certainly, one approaches the speed of light as measured by a stationary observer the closer one gets to the event horizon. At the horizon itself, one cannot emit signals back to the outside world, either.
This is not a particularly productive or useful way to think about falling into black holes, though.
An important point to avoiding the "speed of light" issue is that stationary observers do not exist at or inside the event horizon of the black hole. The velocity of one infalling observer into a black hole will always be less than or equal to the speed of light relative to any other infalling observer. You will never see an example in which two observers at the same point in space have a relative velocity greater than 'c'.
The argument that one is "exceeding the speed of light" requires one to make the false assumption that stationary observers exist inside the event horizon of a black hole. They don't. The "greater than c" velocities occur relative to non-existent stationary observers, they do not exist between any physically existing observers.
I wonder if the OP has possibly been talking to Zanket, who has similar issues.
It's also definitely wrong to think that one's mass becomes infinite when one crosses the event horizon of a black hole, for instance. In one's own frame, one's mass is always constant. In the frame of an observer at infinty, the mass of a black hole will increase by the "energy at infinity" of an infalling object, which is a finite amount, not by an infinite amount.
Thus if a 1kg object falls into a black hole at rest from infinity, the masss of the black hole will increase by 1kg - slightly less, if the infalling object emits any gravitational radiation while it falls in. See for instance MTW's gravitation, pg 904, and /or the discussion of "the first law of black hole dynamics".
While this will undoubtedly be too advanced a book for the OP (original poster) to understand in detail, the fact that the OP is claiming a greater understanding of GR than a graduate level textbook MAY, if we are fortunate, incite a sense in the OP that perhaps there are aspects to GR he does not currently understand. I will not dwell on the less fortunate outomes.
I have to say that I feel I may have been a bit harsh here, but I get concerned when I see people like the OP here get off on the wrong track and not listen to explanations of why they are getting off on the wrong track (the OP was not listening much to SelfAdjoint, either). By expressing my objections a bit hashly I'm hoping to "wake him up" a bit to the magnitude of the problem he's attempting to solve, and the fact that yes, other people have thought about these issues, using much more advanced tools, and that it would be a good idea to study the literature more, and be a bit humble, and make a lot of effort to understand.
This only scratches the surface of the "black hole" issue, BTW.
A much better way to think about the space-time geometry of a black hole is that space becomes so twisted that the radial direction pointing nto the black hole becomes timelike.
This immediately shows why stationary observers don't exist inside the event horizon - we are asking observers to stop time when we demand that they keep their 'r' coordinate constant.
Using this geometric approach to the problem, there are no "forces" to worry about at all. Rather than experiencing a "force", objects simply follow a geodesic path in space-time. One still does run into some unpleasant coordinate singularities at the event horizon, but these can be eliminated with the correct choice of coordinates.
This is the approach that real GR uses (real GR as opposed to the popularized version) - but it takes quite a bit of mathematical sophistication to pull it off :-(.
