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.
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.