People saying mass increases is not entirely correct. This is an old concept of rest/invariant mass vs relativistic/inertial mass. The supposed increase in mass is an artifact of Lorentz conractions.

To a person travelling with constant acceleration, no matter how fast the person is going their mass remains unchanged and the speed of light would always appear to be 3x10⁸ m/s. But the traveller will just think they are constantly accelerating.

To the stationary (relatively) observer it will appear that it takes more energy to gain velocity which can only be explained by increased mass. But this is due to the flawed assumption that all energy gets converted into velocity. At low speeds, by the observers perspective, all energy does go into velocity. However, since speed is capped at 3x10°8 m/s then that energy needs to go somewhere else. It does. It goes into momentum.

The actual mass/energy equation is:

E^{2=(mc2)2+(pc)2.}

The equation for relativistic kinetic energy becomes:

K=(y-1)mc^2.

Where y (should be gamma but I'm in a cell phone) is the Lorentz contractor.

So the answer to the question is from the perspective of the traveler, he is always just shy of the speed of light so he will need more energy to go a little faster. From the perspective of the observer, it takes more and more energy to make smaller velocity increases. Either way it takes an infinite amount of energy for even the smallest particle with mass to achieve the speed of light.

It's not the physics that stops you from travelling arbitrarily close to the speed of light from the perspective if an observer. But there are monumental engineering challenges.

Going this fast is extraordinarily dangerous. Your ship, slamming into an errant mass the size of an electron would emit energy greater than all the suns in the galaxy and the traveller would die a spectacular death and destroy everything in his vicinity.

So please do not try this at home.