If you could constantly accelerate at the rate of 1g you would theoretically be travelling at the speed of light in about 1 year. Could you then just keep accelerating at that rate without ever cracking the speed of light?Assuming you had a ship that maintained a proper acceleration of 1g. It starts at rest with respect a buoy floating in space.
After 1 year, as measured by the ship, it would be moving at 0.77c relative to the buoy.
After 2 years, it would be moving at 0.97c relative to the buoy
after 5 years, 0.99993c
8 years, 0.9999998c
12 years, 0.99999999996c
For someone at rest with respect to the buoy:
after 1 year you would be moving at 0.72c
2 years, 0.9c
5 years, 0.9816954499c
8 years, 0.9927286443c
12 years, 0.9967486193c
In neither case would the ship velocity relative to the buoy follow a strictly v=at pattern, though at the very begin it would be very close. ( for example if you just consider the first 10 days of buoy time, the difference only works out to be ~0.04%)