It certainly going faster each time relative to the speed of light, which increases the mass. At first this is minuscule, but it starts to add up by the time yo

It certainly going faster each time relative to the speed of light, which increases the mass. At first this is minuscule, but it starts to add up by the time you're going a decent fraction of c. It will asymptotically approach infinity, which is why nothing with mass can be accelerated to c. It also means that at some point, you won't have enough energy to continue accelerating.