Long-Life High Altitude Balloon Normally high-altitude balloon experiments end with the balloon popping and the payload falling back down to be reclaimed. 
But if a second balloon was attached to the payload, one which was only partially inflated at launch, then could you keep the balloon aloft for a very long period of time? A sort of extremely-cheap very-low-orbit satellite.
And if so, then does anyone do this?
 A: As a completely engineering / practical statement, you wouldn't let the first balloon burst before the second takes over.  It would be better to use some more gentle engineered systems to transition to different stages designed for different altitude ranges.
But provided you did this, could you continuously increase in altitude until you fall into the formal definition of space?  Yes, this is the exact principle behind orbital airship proposals.
http://en.wikipedia.org/wiki/Orbital_airship
They actually extend the idea further than what you've taken it.  At high altitudes, buoyancy forces are combined with lift (which is an airship by definition) and even orbital acceleration to maintain the craft's altitude.
For such high altitudes, you have no choice other than to use $H_2$ gas, as opposed to the favored $He$ for low-altitude airships today.  But to the underlying physics problem, yes, what you describe is entirely possible.  The gas volume of the later stages does expand as it gets higher too.
A: And what stops the second balloon rising to the height at which it bursts?
High altitude weather balloons are only partly inflated at launch, they rise until the outside pressure is low enough that they pop, or they have vents that release gas to maintain pressure.  The life is normally set by the power to the instruments.
A: There are two limitations to your proposal: the volume of the balloon will be very large at higher altitudes and the gas such as hydrogen and helium will leak even through solid walls with time.
