You ever feel like paleontologists are going to go extinct, like the dinosaurs they hunt for, just because... they found them all?
Short answer, it's physically impossible, but if you could actually get to the speed of light, the universe would end before you could check. Speed of light = infinite time dialation; time of the outside universe would essentially stop and the universe would appear to end quicker than that toy submarine imploded on the way to the Titanic.