10^10^28 Meters

"Even though the whole universe may be infinite we can only ever have contact with a finite part of it because the speed of light is finite . . . There are huge distances but they pale into insignificance when compared with the distance you would have to travel in order to have a good chance of encountering a double. To have a good chance of running into your own double you would need to look out to 10^10^28 meters and for a copy of the whole Earth and everything on it you would need to go to 10^10^30 meters. The first copy of our entire visible universe would not be found with 50:50 chance until we looked out to 10^10^120 meters. The finiteness of the speed of light protects us from any encounters with our doubles in an infinite universe. Yet there is still something deeply disturbing about their existence despite our insulation from encountering them."

-The Book of Universes. John D. Barrow.