Iirc to get to Proxima Centauri at a constant 1g would require a fairly impressive mass of antimatter, to continue to the edge of the observable universe would be so many more orders of magnitude more energy than that it's barely worth discussing. Even though we can't reach c, as we start to approach it the energy requirements rise catastrophically, along with the risk of colliding with some random mote of Hydrogen and remembering that your relative velocity is .99999c, in the moment before you and your ship are atomized.