What's the farthest one can run (say, at 8 miles per hour) in no time at all?
If by time we mean "elapsed time" then obviously the answer is zero. On the
other hand, suppose we measure time according to the local time at wherever
we happen to be. Then it's quite possible to cover a considerable distance
in no "time", or even in a negative amount of "time".

For example, suppose we start a run at Edgerton, Ohio, and finish at Butler,
Indiana -- a distance of about 5 miles. We begin our run at 8:00 a.m. and 
finish at around 7:40 a.m. local time. We finished our run some 20 minutes 
before we started! The reason, of course, is that during the run we crossed 
from the Eastern to the Central time zones, losing 1 hour of local time 
along the way.  (Actually, Indiana does not observe Daylight Savings Time, 
so this example must be modified when DST is in effect.)

Reasoning this way, a quick answer to the question posed above would be
"8 miles", since if we started our run exactly 8 miles East of a time zone
boundary, and the run lasted for one hour, we would finish just as we crossed
into the new time zone at a local time exactly equal to the time we started.
Run any further and it would take some small, but positive, amount of time.

Like most quick answers, this one is wrong. It's not hard to see that a good
ultrarunner could cover about 184 miles in no time at all! To see why, it's
necessary to review some basic facts about time zones. The Earth rotates from
West to East, making the sun appear to move from East to West. Since it takes
the Earth 1 day to complete a full 360 degree rotation, it rotates 15 degrees
each hour. (15 = 360/24.) If point B is located 15 degrees in longitude
West of point A, it will take the sun about one hour longer to reach the
same position in the sky as seen from point B as it does when seen from point
A. Without time zones, "noon" on the local clock would not generally be the
same as "astronomical noon," the time of day when the sun is directly 
overhead. With time zones, the local time at point B is one hour earlier
than at A, and the sun would appear to be at about the same point in the sky
at a given local time from both locations.

There are 24 time zones whose boundaries roughly follow meridians of longitude.
Greenwich, England, is in the middle of a time zone, and each zone spans 15
degrees of longitude.  Thus a location at 75 degrees West longitude would
be in the sixth time zone (counting the zone of Greenwich as zone 1,) and so 
the local time there would be 5 hours earlier than at Greenwich. (In the 
populated temperate zones the time zone boundaries don't exactly follow 
meridians of longitude due to political considerations, but they do so on 
average. Near the poles the time zone boundaries coincide with meridians of 
longitude.) 

A supersonic jet flying from Paris to New York can easily land before it
took off because it covers the distance from one time zone to the next in less
than one hour. The distance from one time zone to the next along the 45th
parallel of latitude is about 700 miles, so a runner could not hope to 
duplicate this feat in the temperate regions. But what about near the poles?
Near the poles the distance between time zones becomes negligible. For example,
if you walk in a tight circle around the North pole, it's easy to span
entire time zones in a single step!

Thus a runner, beginning some distance south of the North pole and
running westward, could easily cross entire timezones in less than an hour,
effectively covering more and more distance in less and less time. 
Unfortunately, the party ends at the International Date Line (180 degrees
longitude,) where the date abruptly changes to one day later. 

So here's a way an ultra runner could cover 184 miles in no time at all. Begin
just West of the international date line at a point about 30.5
miles due South of the North pole (or due North of the South pole) and run
directly towards the West at 8 mph. The run ends just under 24 hours later at 
a point East of the dateline and at the same local time it started. 
The distance covered is 23/24th of the circumference of a small circle of 
radius 30.5 miles, or about 184 miles if we ignore the small correction 
due to the curvature of the Earth. (The reason it is 23/24th of the
circumference and not the entire circumference is that the runner must stop
immediately upon entering the last time zone prior to returning to the
date line.)

Is it possible to cover even more than 184 miles, perhaps taking advantage
of certain special circumstances? I'll leave this as a puzzle. (Note: you
must run on the surface of the earth itself -- no running along the aisles
of airplanes, e.g.)