Free fall is the key way of describing it. A satellite in orbit is always falling towards the Earth, at the standard rate at which any object falls toward the Earth from that height. It's just going fast enough in a vector, as Nick pointed out, transverse to "down" that the Earth's curved surface falls away from the satellite as fast as the satellite falls toward it.

-the other Doug

There are balloons that can go higher, but they tend to be much larger, and much much MUCH more expensive. If you're lucky and have a small payload, a typical met-balloon might reach 120,000ft, but not significantly more than that.

Altitude

Gravity decreases with altitude, since greater altitude means greater distance from the Earth's centre. All other things being equal, an increase in altitude from sea level to the top of Mount Everest (8,850 metres) causes a weight decrease of about 0.28%. (An additional factor affecting apparent weight is the decrease in air density at altitude, which lessens an object's buoyancy.[3]) It is a common misconception that astronauts in orbit are weightless because they have flown high enough to "escape" the Earth's gravity. In fact, at an altitude of 400 kilometres (250 miles), equivalent to a typical orbit of the Space Shuttle, gravity is still nearly 90% as strong as at the Earth's surface, and weightlessness actually occurs because orbiting objects are in free-fall.

The balloon grows as the atmospheric pressure outside decreases. This maintains it's buoyancy - but eventually it just expands so much, the rubber/latex can't cope and it breaks. For balloons of this type, that typically happens at about 100,000 ft.