How high does an average limp person have to fall from to hit the
ground at terminal velocity?  Honestly, I'm only asking this question
out of pure curiosity, and so I can point out to acrophobic friends
that being on a balcony on floor x is no more life-threatening than
being on a balcony at floor y.  I'd like to know what the minimum
value of y is for this statement to be true.

Pomakis..

The terminal velocity of a falling body occurs during free fall when a
falling body experiences zero acceleration.  The magnitude of terminal
velocity depends on the weight of the falling body. For a heavy
object, the terminal velocity is generally greater than a light
object. This is because air resistance is proportional to the falling
body's velocity squared.  There is much more to determining terminal
velocity than simply body posture and height.  There are, however,
some rough estimates that you can use in order to prove your point to
your friends.

It is estimated that the human body will reach 99% of terminal
velocity after falling 1,880 feet (573 meters) which takes anywhere
from 10-14 seconds.  With normal posture and normal atmospheric
pressure, this is a speed between 117 and 125 miles per hour, or
approximately 54 meters per second.  Another source I found stated
that a human would have to fall over 400 yards (1,200 feet) prior to
achieving terminal velocity.

Now, a building story is approximately 12.5 feet.  Using the lower
number of 1,200 feet and the very basic information above, you would
have to fall from approximately the 96th floor balcony to achieve
terminal velocity.  Now, I am sure there will be a physics genius or
two that will calculate the exact distance and speed required using in
depth formulas, however, I feel that the above information (and the
references below) should suffice for your friends. :)

Thanks for you very interesting question and I hope the above
information is helpful.  If you need any additional clarification,
please let me know.

Regards,

-THV

Search Strategy:
terminal velocity human

References:

The AFU and Urban Legend Archive Death - Falling Terminal Velocity
http://www.urbanlegends.com/death/falling_terminal_velocity.html

Speed of a Skydiver (Terminal Velocity)
http://hypertextbook.com/facts/JianHuang.shtml

Ask a Scientest - Falling Bodies
http://newton.dep.anl.gov/askasci/phy00/phy00012.htm

Physics Forum
http://www.physicsforums.com/topic.asp?TOPIC_ID=7318

There is no way anyone could fall 573 meters in 10 seconds, because
even in a vacuum it would take over 10.8 seconds to fall that far. 
With zero air resistance, you would only fall 490 meters in 10
seconds.

Just to note - terminal velocity isn't the measure of serious harm to
a falling human body. As pointed out, terminal velocity is the point
at which air resistance counteracts gravity's acceleration.

Falling from higher than 30 ft. is potentially fatal. At 30 ft.,
you'll hit the ground at approximately 22 mph. And of course, higher =
faster.

Even a fall from 10 ft. can cause serious injuries. From that height,
you're only going to hit the ground at 8 miles an hour (a brisk
jogging speed) but would you jog straight into a wall?

Without air resistance, falling from 10 ft means you hit the ground at
17 mph, and falling from 30 ft means you hit the ground at 30 mph. 
Air resistance at these speeds cannot account for the differences
between these numbers and flajason's (especially the 10 ft value).

You are correct racecar. I neglected to use the correct formula for
calculating the speed at impact. I was solving for the average
velocity over the course of the fall, not the speed at impact.

My apologies if I provided any misleading information.

Having re-worked the math:
30 ft. = 29.67 mph
10 ft. = 17.24 mph

-Flajason