This description doesn't seem quite right to me. The laws of physics work the same in all reference frames, so from the frame of the crew, day 355 would simply

This description doesn't seem quite right to me. The laws of physics work the same in all reference frames, so from the frame of the crew, day 355 would simply be yet another day of 1G acceleration. At least inside the spacecraft, they would notice no difference from day 354 (or any other previous day). When they look out the windows, they will see odd things with respect to the apparent speed of other visible objects (including they never get farther/closer at a rate faster than C). But there's nothing theoretical that prevents you from continuing to accelerate at 1G forever. Edit: I guess what I don't like about your description is that you mix reference frames. A spaceship can maintain 9.8m/s^2 (from its perspective) infinitely with constant energy expenditure. Energy expenditure rises asymptotically only if you require constant acceleration of the spaceship from Earth's perspective. But that's a peculiar way of framing it.