What is the maximum altitude that a helicopter can achieve?
It varies with the type and design.
The answer is complicated because of the large variety of helicopters in the world.
The BELL 206 light helicopter was limited in a hover to around 8000 ft. If it tried to hover above this altitude, the tail rotor would basically stall out.
Also, a helicopter (and an airplane) flies more efficiently close to the ground, which is referred to as "ground effect". For an airplane, the height above the ground is about 1/2 the wing span for it to receive additional lift effect due to in ground effect. A helicopter would be about the same: 1/2 the rotor diameter. So, a helicopter might be able to hover 10 feet above the surface of an 8000 ft mountain.
On May 14th, 2005, an Ecureuil/AStar AS 350 B3 helicopter operated by Eurocopter was reported to have landed on Mount Everest (29,035 feet). The landing is in dispute. The listed service ceiling of the rotorcraft is between 17 and 18 thousand feet, which is considerably short of the summit altitude.
An unmanned high altitude helicopter is nearing completion. TGR Helicorp in New Zeland has designed the "Alpine Wasp" specifically for rescue evolutions on Everest. The machine's diesel engine will give it an operating ceiling in excess of 30,000 feet.
The CH-47 Chinook twin rotor helicopter is used by the USAF to rescue climbers on Mount Denali (McKinley) in AK. It can reach an altitude in excess of 19000 to land at an elevation of around 18000. The biggest problem at that point is restarting the engines, so a special storage device directs pure oxygen into the engine inlet to restart.