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I think the other posters are forgetting that nature allows for much more possibilities. True, you can't catch up to light, but that might not be necessary. As General Relativity tells us, space-time is a 4 dimensional manifold. Einstein's equations tie its local structure to the energy-momentum density. But that still leaves open a lot of possibilities for the global structure of the universe.

Imagine that the universe would be something like a torus or a sphere, that is finite but unbounded. This concretely means that if you go long in enough in one direction, you will ultimately end up where you started again. Now, that is also true for light. If then the expansion of the universe did not happen so fast that light going around got behind the cosmological horizon, it is theoretically possible that we could see our own galaxy in an earlier stage of its evolution.

The global shape of the universe is still an open problem. The most commonly accepted model though assumes an infinite flat space-time. Other models like Poincaré dodecahedral space and the Picard horn do seem to fit current cosmological data. Well, they did last time I looked, maybe they have been refuted by now.

But if the universe is well described by a Poincaré dodecahedral space FLRW model, then there is a chance that we can see our own past for a limited amount of time, i.e. until it disappears beyond the cosmological horizon. Don't hope to see yourself though.