Is plane altitude limited by engine power and if so does air density cause this? I notice that, for example, human-powered flight operates at low altitudes. This might of course be due to safety but I wonder if in fact the delta in air pressure is greater at lower altitudes and this prevents low-powered aircraft from reaching higher altitudes?
 A: Human-powered flight is only barely possible because humans cannot generate large amounts of power. Human-powered planes therefore fly almost exclusively in what is called ground effect, where the plane is no more than about one wingspan off the ground. When flying this close to the ground, the plane is partly supported by a "bubble" of air that is caught between the underside of the wing and the ground, which reduces the amount of drag experienced by the plane and enhances the lifting power of its wings. This in turn reduces the power requirement to keep the plane in the air. 
Regarding small planes with low-powered (piston) engines, they are limited in how high they can fly because as they climb they move into air which is less dense and which therefore furnishes less oxygen for the engine to burn with its fuel. This causes the power output of the engine to decrease with increasing altitude, and at some point the engine's power output is so diminished that the plane can climb no further. 
In small planes, this maximum altitude point can be increased by putting a bigger engine in the plane, reducing the plane's weight, or putting a compressor on the engine's intake to furnish it with denser air at high altitudes so it can deliver more power. 
In large planes with jet engines, those engines are specifically designed to deliver high power and best economy at high altitudes.
A: Here's what happens when airplanes go to higher altitudes.
First, there's less oxygen, so they have to have ways to get enough, like turbo-charging. Jet engines have built-in turbo-charging, and supersonic engines also use ram effect.
Second, since the air is less dense, they need to go faster to get the same lift. That's a good thing. It's why long-distance aircraft go as high as they do. But there's a problem. In colder air the speed of sound comes down, and if their airspeed approaches the speed of sound they can get into problems. This is called the "Q-corner" or "coffin-corner". To go faster than that they need to be designed for supersonic flight. So, for example, subsonic transports are limited to about 40k feet, while the supersonic Concorde cruised at 60k feet.
For human-powered flight, it's strictly a matter of having enough power to overcome drag. Niels is right about ground effect. When an aircraft is within one wing-length above the surface, drag is greatly reduced, so much less power is needed. If you take a flying lesson you learn to watch out for this, because you don't want your aircraft to "float along" when you're trying to get it on the ground. But it can help if you're taking off on a poor surface. You get into ground effect and stay there while you accelerate.
