What is the maximum altitude where humans can survive?
Short answer
Between 62,000 and 63,500 feet (18,900 and 19,350 meters) blood begins to boil at body temperature. This altitude, referred to as the Armstrong limit, is generally considered to be the absolute limit compatible with life. At this point, humans cannot survive without pressurization measures.  

Background
Atmospheric pressure drops at higher altitudes. As you go higher into the atmosphere, gases start to expand due to the lower atmospheric pressure. This is what causes your ears to pop when driving up in the mountains or when an airjet lifts off. At altitudes higher than 10,000 feet (3.5 km) people unaccustomed to the reduced atmospheric pressure may become short of breath or experience dizziness. At altitudes above 20,000 ft (6.1 km) supplemental oxygen is required to support life and at 34,000 ft (10.4 km), 100% oxygen is necessary to equal the partial pressure of oxygen in sea level air. Above 40,000 ft (12.2 km), 100% oxygen must be administered pressurized (NASA).

Altitudes above 50,000 ft (15.2 km) are considered near-space and man requires a pressurized suit to be safe. At 55,000 ft (16.8 km), atmospheric pressure is so low that water vapor in the body starts to boil at ambient temperatures, causing the skin to inflate. At 63,000 ft (19.2 km) blood at normal body temperature (98 F / 37oC) starts to boil. At altitudes above 65,000 ft atmospheric pressure approaches that of space, i.e., a vacuum (NASA).

Between 62,000 and 63,500 ft (18.9 - 19.4 km), blood at body temperature (37oC) begins to boil. This is the point beyond which humans absolutely cannot survive in an unpressurized environment. This point is named the Armstrong limit, after Harry George Armstrong, who founded the U.S. Air Force’s Department of Space Medicine in 1947. On Earth, the Armstrong limit is somewhere between 62,000 and 63,500 feet (18.9 and 19,4 km), or about 12 miles.

As a reference, the highest point on earth is the peak of the Mt. Everest at 29,000 ft (8,8 km). 

Above the Armstrong limit, water in the soft tissues of your body vaporizes, causing gross swelling, though the tight seal of your skin would prevent you from actually bursting apart. Your eyes, likewise, would refrain from exploding, but continued escape of gas and water vapor leads to rapid cooling of the mouth and airways (Goslin, 2008).

Water and dissolved gas in the blood forms bubbles in the major veins, which travel throughout the circulatory system and block blood flow. After about one minute circulation effectively stops. The lack of oxygen to the brain renders you unconscious in less than 15 seconds, eventually killing you. When the pressure gets very low there is just not enough oxygen, which is thought to be the first and most important concern (Goslin, 2008). 

Note that under conditions of near-vacuum, death is not instantaneous. At one three-hundred-eightieth of atmospheric pressure at sea level, dogs were shown to survive up to 90 seconds. During their exposure they weren't in a great state (the dogs were bloated, ice covered their tongue, and many more horrible facts to be read in (Goslin, 2008)). Monkeys were even shown to cope for 3.5 minutes under these conditions.

Reference
- Goslin, Sci Am February 2008