I believe this could only happen around a black hole. In that case, yes: light that we emitted umpty-million years ago could be shot back at us. The problem is that there would be no focussing. At best it would be like looking at an Earth 2 x umpty-million light years away. I'm guessing that it would actually be worse, with the black hole dispersing light.
(IANAastronomer, but I have opinions on any given topic...)