According to the theory of Special Relativity, it would take infinite energy to get to exactly light speed, but particles can be accelerated to almost the speed of light, and their momentum measured directly, and the mass increase derived from the measurement.
If the propellant is on board with the object, then it's mass increases with the object and tends to infinity; it should be sufficient to provide the infinite energy source. But, back to your earlier point, I believe there is no proof of this ever happening.
Particle accelerators have to take relativity into account as particles approach the speed of light. No one can accelerate a particle to the speed of light, but the energy required to accelerate a particle to near the speed of light is comparable to the energy used by a city, due to the increase in mass.