Since the exosphere has no specific end, and it is almost into orbit, then yes. When a space shuttle reaches orbital velocity, it is going to fly past the exosphere. A space shuttle has never attained escape velocity that I know of. It is certainly capable, though.
Also, a space shuttle can be flown in the exosphere without ever reaching orbit in some conditions. (i.e. if they launch at JFK in Florida and land at Edwards Air Base in California, because the earth moves counterclockwise. They could only be up there for a very short time, however.)
the short answer is the space shuttle Columbia on STS-1
As of the end of February 2010, NASA has flown 130 space shuttle flights, two of them ending in catastrophe.
Neil Armstrong has never flown on a space shuttle mission.
exosphere
In the exosphere, Friction between the atmosphere and the shuttle creates the heat.
the short answer is the space shuttle Columbia on STS-1
The space shuttle Discovery has flown the most times of all the space shuttle orbiters.
the exosphere.
the challenger
Neil Armstrong has not flown on the Space Shuttle.
As of the end of February 2010, NASA has flown 130 space shuttle flights, two of them ending in catastrophe.
Neil Armstrong has never flown on a space shuttle mission.
exosphere
exosphere
Columbia
2011
In the exosphere, Friction between the atmosphere and the shuttle creates the heat.