Assuming the universe is curved and that is it possible to travel in a straight line and return to your starting point, it would make sense that at a younger age when the universe was smaller that light from an object sent in opposite directions would eventually reach the opposite "pole". Do we see any possibilities of this? Seeing the same object from opposite sides in opposite sides of the sky? Such object would have to be very old, I know.
We can't, because the universe isn't curved enough to allow light to travel far enough since the big bang
If the universe was more highly curved, this might be possible. And the objects that appeared "twice" might not be very distant - or only one of the images would be distant. It was even speculated that it might be possible to see the milky way and Andromeda as a distant pair of galaxies.
There were also searches of the cosmic microwave background to investigate if parts of it could be different views of the same pattern. These turned up negative.
So while "yes" if the universe had sufficient positive curvature, it might be possible to see the same object twice, it turns out that the curvature of the universe is very close to zero, so we don't.