I will answer this one with two catagories, where you live afterwards and where you don't. For surviving, I think the fastest a human has ever gone with no mechanical assitance is about 26 mph in a running sprint. For not surviving, the human body in freefall can reach speeds of around 120 mph before being limited by terminal velocity. However, if you had no machines involved there would be no parachute, so you would not survive the experience.
This answer is wrong. Speed skating is non-mechanical and people skating can reach speeds of over 30 miles per hour. Skates are just blades attached to your feet and are not "mechanical". We could call them aids but they are definitely non-mechanical. Japanese Joji Kato is the world record holder for the 500 meter sprint where he averaged 14.58 meters/second which is 32.39 mph.
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The fastest speed ever achieved by a human without a machine is around 27.8 miles per hour (44.7 km/h) by Usain Bolt during his 100-meter sprint world record in 2009.