As Sam Stone said, there are two overing ceilings for helicopters: In ground efect and out of ground effect. Ground effect is a “cushion” of air that helps support aircraft (including fixed-wing) within about a wing span from the surface (or is it half a wing span?). A wing flies more efficiently in ground effect, so they can fly at a higher altitude IGE. The IGE hovering altitude of most helicopters is well below the altitude of the summit of Chomolungma.
Another factor with helicopters is retreating blade stall. In order to generate lift, a wing must have a certain amount of air flowing over it. When the airspeed slows the angle of attack must be increased to create the same amount of lift – up to a point. At a critical angle of attack the air won’t “stick” to the wing and the wing stalls. This can happen at any airspeed and at any angle of attack. (Don’t worry, it’s not capricious.)
Now consider the helicopter rotor system. At hover and in no wind, the rotor blades have the same airspeed; say, 300 miles per hour for the sake of illustration. If the helicopter flies at an airspeed of 100 miles per hour, the advancing blade has an airspeed of 400 miles per hour and the retreating blade has an airspeed of 200 miles per hour. If the helicopter flies fast enough the airspeed of the retreating blade slows to a point where it can no longer develop lift. The rotor system blows back, chops off the tail boom, and the crew and passengers plummet. Or maybe there is no boom chop, but they plummet anyway. I’ve seen video.
We all know that atmospheric density decreases as altitude increases. An airplane can fly at a higher speed to create the same amount of lift because both wings have the same airspeed. (Indicated airspeed will be lower than the true speed of the aircraft.) Helicopters can’t do this because of their retreating blades. To compensate for altitude they must slow down so there is enough air flowing over the retreating blade to prevent stall. At a very high altitude even a slight breeze may be enough to stall a blade. Consider that stall speed and the top speed of the TR-1 (U-2) surveilance aircraft are seperated by only about five knots.
Assuming a helicopter could reach the summit of Everest, and assuming it has enough power to actually hover, and assuming that there is zero wind, a helicopter might be able to land on the summit. But there aren’t any that have the power to do that. Perhaps a helicopter could be purpose-built for such an experiment? Maybe with extremely long blades? That will take more power. More power? That means more fuel to feed the bigger, thirstier engine. More fuel? That means reduced paylod. Solution? Bigger engine. Which needs more fuel, etc. I suppose it could be done, but why?
An anecdote: Filmmaker/skiier (dang, I forgot his given name) Miller took a helicopter to a slope in New Zealand many years ago. IIRC it was a Bell 47, a three-seat piston-powered aircraft. It could have been two seats, as IIRC one of the skiiers had to be outside on a litter. The helicopter didn’t have enough power to take off with such a heavy load, so it made short hops down the slope until it was low enough to fly off of a precipice. It fell quite a long way before there was enough air density for it to sustain level flight. My memory is fuzzy on this anecdote, but that’s how I remember it.