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The amount by which a spacecraft is able to change its velocity is called it's Δv (delta-velocity) budget. You can calculate the Δv-budget of each stage of a rocket using the Tsiolkovsky rocket equation which reads: $$ \Delta v = I_{sp} * 9.81 * \ln \frac {Mass_{full}} {Mass_{dry}} $$

where I sp is the specific impulse ("fuel-efficiency") of the engine. The factor of 9.81 (the gravity of earth) in this context is the factor used to convert specific impulse to exhaust velocity. You could also express the formula above using exhaust velocity, but I decided to use the specific impulse, because the numbers I found for the efficiency of the Saturn V were in I sp .

The masses and I sp of each stage can be looked up on wikipedia. I made an overview as a table and calculated the delta-v using this handy online calculator:

| Individual stage | Total vessel | Stage | Full mass | Dry mass| Full mass | Dry mass | Isp | Δv ------+-----------+---------+-----------+----------+-------+-------------- I | 2,300,000 | 131,000 | 2,900,000 | 731,000 | 263s | 3554.2 m/s II | 480,000 | 36,000 | 600,000 | 156,000 | 421s | 5561.5 m/s III | 120,800 | 10,000 | 120,800 | 10,000 | 421s | 8796.2 m/s =========== 17911.9 m/s * masses are in kg

That means a Saturn V started in space could reach a speed of about 18 km per second or 64,500 km/h relative to its initial frame of reference. The speed of light is about 300,000 km per second, so this is still just about 0.006% of the speed of light.

This speed is just the rocket without any payload. When you add a payload, that mass needs to be added to the dry- and full-mass of the total vessel in each stage and the delta-v of each stage decreases accordingly. I didn't do the math, but when you would add the Apollo 11 payload, the 12 km/s stated in the answer by Mark Adler appears to be plausible (but note that the Apollo 11 payload would add three additional stages with additional Δv budget - the service module, the landing module descent stage and the landing module ascent stage).

Note that the first two stages of the Saturn V were optimized for operating in the atmosphere of Earth, not for the vacuum of space. When you would build a rocket only for operating in vacuum, you could expect each stage to be as efficient as the 3rd of the Saturn V. Without the acceleration requirement during launch, you wouldn't need such large and heavy engines like those on the first stages of the Saturn V. You would reach a much higher Δv by having only a single small engine and dropping only the empty fuel tanks themself. But that still would not nearly be enough to get close to the speed of light.

Another interesting trivia fact: Incidentally, 18 km per second is quite close to the current speed of Voyager I. Note that Voyager I was launched with a rocket much smaller than a Saturn V (a Titan IIIE). How could it still reach such a high speed? By performing multiple gravity slingshots around the various planets it flew by. What conclusion can we take from that? When you want speed, don't just add more power. Use the power you have in a smarter way.