The previous answers give an excellent overview of limiting factors regarding high altitude flight with powered civil aviation aircraft like a small Cessna.
Of interest might be that people have operationally used these aircraft at altitudes in excess of the aircraft's original-design flight envelope. One use in particular which comes to mind has been in special applications such as high-altitude aerial-photo surveys. A modified Cessna Turbo Stationair has been used in such high-altitude aerial surveys and was flown at altitudes above 37,000 ft (11.3 km). The plane's engine, modified to have a two-stage turbocharger, could maintain sufficient manifold pressure to fly at such altitudes.
The interior of the plane was essentially stripped of unused, excess-weight items (five of six seats) and other miscellaneous and unnecessary equipment. The camera bay, aerial camera and camera operating equipment, necessary flight instrumentation and radios, and sufficient oxygen (tanks), were items carried on such flights, along with required fuel. The owner of this plane commented that at such altitudes, the plane did not fly especially well, but it was manageable. He also commented on the interesting view looking down on commercial jet aircraft flying at altitude below.
Also of interest might be that other much lighter, essentially smaller, commercially built, single-place, unpowered aircraft have been routinely flown at altitudes well above 40,000 ft (12.2 km). These are sailplanes, and oxygen is required on such flights. Although generally less than 600 lb (270 kg) in weight (one-fourth the weight of a small Cessna), they have larger span wings (55 ft or about 17 m, more or less) with generally similar wing area compared to a small Cessna powered aircraft. Paul Bickle, retired director of the NASA Dryden Flight Research Facility, flew a Schweizer 1-23E sailplane to an altitude of 46,267 ft (14.102 km) on February 25, 1961. His flight was within a Sierra wave and was just over 2 hours from take-off to landing. Bickle's record for altitude gained, 42,303 ft (12.894 km), is unchallenged in a single-place glider. He suffered outside air temperatures of -65 deg C which frosted the inside of his canopy so badly he could not see the expansive view before him. He only had his instruments to guide his flight, and, because he could not close his cockpit outside-air vents, became so chilled and distracted by the cold that he had difficulty paying attention to piloting his sailplane. Of course, he was more attentive as his canopy cleared in warmer air at lower altitude. Also, Robert Harris flew a single-place Grob 102 sailplane to a record altitude of 49,009 ft (14.94 km) on February 17, 1986. His flight was also in a Sierra wave.
In 1952, Larry Edgar and Harold Klieforth set an altitude record in a two-place Pratt-Read G-1 glider, soaring to 44,255 ft (13.489 km) in a Sierra wave. However, on April 25, 1955, Larry Edgar's Pratt-Read glider was destroyed in the lee of the Sierra by a rotor-cloud at 17,000 ft (5.2 km) as he was investigating the rotor's turbulent structure at the base of a wave. The acceleration he experienced, in excess of -20g, ripped off his helmet, boots, gloves, and oxygen mask. As he drifted downward he could see parts of his glider being carried upward and worried if he pulled his parachute rip-cord, he might be carried upward as well. Fortunately, he was able to make a parachute landing and survived without breaking any bones. The extreme negative acceleration partially damaged his vision. Edgar was the only one in his sailplane when this happened.
As one can see, all caution must be used when flying an aircraft in conditions that are not within the original aerodynamic-design flight envelope of the aircraft. The plane will fly, but nevertheless, aerodynamic control is critical. This brings to mind the following caution that is especially true in high-altitude flight, and known all too well...
Aviation, although not inherently dangerous, is, to an even greater extent than the sea, terribly unforgiving of any carelessness, incapacity, or neglect.