I think you're missing the point. It isn't a limitation on how you get the acceleration. You could have an arbitrarily large acceleration; millions of g's, billions of g's if you want. You could achieve it through any physical means your imagination allows; you want a gravity-magnifier? Granted. You want a limitless power source? Granted. You want to keep that acceleration up for a year? a millenium? a million years? Granted.
Your velocity would not reach, let alone exceed, c. Ever. Your velocity would merely get arbitrarily close to c, i.e. .99999999999...c
Velocities near c do not simply add. They follow the formula: v(1) = (1-(v(0)^2 / c^2))^.5.
You will find that, no matter how much you add velocities of any size (i.e. by hard acceleration) or how many times you reiterate the formula (i.e. by accelerating for a long time), your velocity will never exceed c.