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I haven't studied this in a while and do not work in the field, so this might be incorrect or imprecise.
It is not only impossible to accelerate to the speed of light, it is impossible for anything to have a speed higher than the speed of light.
While it is counter intuitive, even relative speed cannot be higher than the speed of light. If you are on a train going 300 km/h and another train is going the opposite direction at 300 km/h, then that other train, from your perspective, is going at (almost) 600 km/h.
But when we get close to c, things change: if in a referential, you are in a spaceship going at 0.9c, and you observe a photon going at c in the opposite direction, from your perspective, that photon is going at c for you, in the spaceship, and it is also going at c for someone stationary in the referential observing from the outside.
Similarly, If you are in that spaceship, and observe a spaceship going in the opposite direction at 0.9c as well, then you won't observe them as going 1.8c in your referential, but something close to yet below 1c. I forgot the formula.
Nothing can be faster than light, in any referential.