What happens if we exert more force on an object which would be travelling at near maximum light speed? If we exert force on an object which would be travelling at near light speed, to an extent to make it move greater than speed of light in vacuum, what would happen? If object doesn't move faster than light, what happens to that excess force?
Is this experiment been done? What would happen if we exert force on light travelling at its maximum speed?
 A: If you apply a force (in the direction of motion) to an object with mass, moving close to the speed of light, the object will accelerate and its speed will become even closer to the speed of light. But no amount of force can make it reach, or exceed, the speed of light.
The reason that objects behave this way is that momentum is not $m\mathbf{v}$ as Newton thought; it is in fact $m\mathbf{v}/\sqrt{1-v^2/c^2}$ as Einstein realized. Applying a constant force will cause the momentum to increase indefinitely, but the momentum becomes arbitrarily large as the speed approaches $c$.
There is no way to exert force on light. It always travels (in vacuum) at the speed of light, and cannot be sped up or slowed down.
Why? Well, photons don’t have charge, so they don’t feel electromagnetic force; in fact they carry electromagnetic force. Nor do they feel the strong or weak nuclear forces. They do feel gravitational force, but it doesn’t make them move faster or slower. (In General Relativity, they just move on lightlike geodesics.) These four forces are all the fundamental forces, as far as we know. 
There is plenty of experimental evidence for these facts.
A: Its speed would get yet  closer to that of light, albeit tending to infinitesimally so as the initial speed tends towards the speed of light; and the mass would increase without bound. It becomes simpler to think of it in terms of putting energy into the particle, rather than applying force to it. You could well say that as the body's motion exchanges the Newtonian regime (speed a small fraction of c) for the ultrarelativistic regime (rest mass energy m₀c² a small fraction of kinetic energy), increase in speed for a given small increase  in momentum (small fraction of m₀c) is exchanged for increase in mass for a given increase in energy (that doesn't have to be small relative to m₀c²). This phenomenon is routinely observed: for physicists working with particle accelerators, it's as routine a phenomenon as vehicles passing along the road is to a city dweller; and also it's why an elementary cyclotron has an upper bound on the energy to which it can accelerate protons of ~42MeV.
