This is really more a question of comfort and safety. For a passenger jet, the highest most will every fly is 45,000 feet. Higher is possible but at such high altitudes pressure in the cabin required to keep passengers comfortable begins to strain on the cabin walls. At higher altitudes stronger cabin walls are required to hold the higher pressure in the cabin. The extra weight of reinforced cabin walls aren’t worth the efficiency achieved by flying higher.
In aircraft like the SR-71 and U-2 these aircraft were not fully pressurized and the pilot wore a pressurized space suit to operate the aircraft in order to save on weight by not pressurizing the cabin. These aircraft could fly up to 85,000 feet. Although this is very high, it is nowhere near the edge of the atmosphere. In fact, this is well within the stratosphere.
The X-15 was able to reach altitudes of 350,000 feet, but these were ballistic zoom climbs and not really flight per se. The runner up for highest flying aircraft is the X-43 Hyper-X (featured in my avatar) which sustained level flight using air breathing SCRAMJET engines at 110,000 feet. If you want to add dirigibles into the equation, these could theoretically reach higher heights than conventional aircraft could due to added buoyancy, but since the question is specific to planes, I’d have to say that 100k feet is probably close to the upper limit for air breathing aerodynamic aircraft.
@Poser Also, the B-52 has a service ceiling of 50,000 feet. This means the aircraft cannot maintain steady level flight above this altitude. 70,000 feet is far above this and would certainly cause extreme fatigue on the airframe (if not cabin pressure failure) due to the pressure differential across the cabin walls.