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Asaf Pe'er

Theoretically, could you go far enough into space to slow down time enough for it to stop?

No.

First, there is a misconception in the question: your location in space does not matter as far as time is concerned. What matters is your speed, relative to other people (observers, to use the technical word). If you move relative to another person, your internal clock slows down. This is true regardless of where you are, or what speed you are - including, e.g., if you go for jogging. Of course, the difference in times when you go on jogging is so tiny it cannot be detected. Once your speed increases, so does the slowing of the clock. In fact, this effect can be detected in airplanes, or satellites. It is important enough that it must be compensated for in systems where excellent accuracy is needed, such as, e.g., the GPS system, which is based on very accurate signal sent from satellites orbiting earth.

The faster you move, the more pronounced the slowing down of the clock is. However, there is a limit to how fast you can move: you can never move faster than light, neither you can move at the speed of light (only particles without mass, such as photons - quanta of light, can move exactly at the speed of light. Everything that has mass can never reach this speed). For particles that travel at the speed of light, time indeed stops; From the point of view of a photon that was emitted, say, in the sun and reached us, it took zero time from emission to absorption. However, since you cannot move at the speed of light, you can make your clock goes slow (relative to others; from your point of view, nothing will change. You could not live forever!), but never stop it.

In order to get a substantial slowing down, your speed must be very close to the speed of light - say, if you move at 80% the speed of light, the slowing down will be by a factor of 5/3 (namely, when someone on earth measures 5 minutes, you will measure only 3 minutes elapsing). The speed of light, though, is extremely high: it is roughly 300,000 km/s. The maximum speed achieved by mankind so far is about 16 km/s (equivalent to ~58,000 km/hour), during the launch of NASA's robotic space craft mission "New Horizons" to the planet Pluto in 2006. This is roughly 0.005% of the speed of light.