Wikipedia has the answer: the shuttle was designed to operate at altitudes between 185 km (115 miles) and 960 km (596.5 miles). The actual altitude it flies for any given mission depends on their target. For example the International Space Station is 180-190 miles up, while the Hubble Space Telescope is 360-370 miles up.
Now that the question is answered, I need to address some big misconceptions here: There IS gravity in space. Gravity is what makes it possible for our astronauts and satellites to stay in orbit; gravity is what keeps the Moon in orbit around the Earth; and gravity is what keeps all the planets in orbit around the Sun. What's important is that there is effectively zero air resistance in space. That way, things in space can travel at speeds that would be ridiculous here on the surface of Earth and they won't be slowed down by friction (except for a very, very small amount in low orbits). See the "orbit" link for more info.
What's important to take away from this is that the space shuttle is limited in altitude because it CAN'T carry infinite fuel. Even if they lighten the orbiter so much that they don't even carry anything in they payload bay, it's not going to go higher than 600 miles up. When they designed the space shuttle, they never meant it to go any higher than altitudes that are within Low Earth Orbit (LEO).
UPDATE: The slow spiral that Billy describes would require continuous acceleration throughout the spiral -- which means continually burning up the fuel. As soon as the fuel cuts off and the acceleration ceases, the shuttle would then be in a new, higher orbit and would no longer be spiraling away. This would be one way to accomplish something called orbit raising. This requires fuel, and thus the max altitude is limited. (A very large amount of acceleration would be needed to achieve escape velocity.)