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Make mine a double.
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Instead of a single strand from earth to orbit you have a series of aerostats (balloons, zeppelins, solar-powered helicopters, morning dew in a jar--whatever) and you link them up only via a series of conveyor magnetic fields which form a mass driver. Please excuse if this is posted here elsewhere!
The important thing is that you don't really need one long cable from earth to orbit or half-way, or wherever. You can do it that way, but it seems overly-complicated. You simply need to have the output from power stations that can lift something to the next highest level at staggered intervals. Hanging a transport system from one cable seems a.) like putting your eggs all in one basket and b.) tempting technology and electrical energy transfers by mother earth to deny construction. For that matter, you could hang the last step or two up as an orbital tether within the very uppermost reaches of the atmosphere if you run out of buoyancy in the upper stratosphere. For that matter, if you had a tether like this, as Dagwood points out, you could bypass the whole mess by building a high-altitude dedicated cargo jet to toss along cargo at or near apogee. I guess the idea waz "virtual space elevator" or "disconnected aerial inductrack" using a series of discrete stations as opposed to the more common one-strand unobtainium (ok with nanotubes notquiteyetobtainium) variety.
Very similar idea already 3/4 baked
Half-a-space-elevator project [cloudface, Oct 04 2004, last modified Oct 21 2004]
Space elevator page
[cloudface, Oct 04 2004, last modified Oct 21 2004]
Similar point of view
This company is, in fact, using balloons to reach space... [cloudface, Oct 04 2004, last modified Oct 21 2004]
Kinda baked, I guess...
Noted aerospace scientist A.A. Bolonkin came up with an idea quite a bit like this in either 2003 or 2007...his inflatable space elevator is one piece, however... [cloudface, Jun 06 2009]
||If I was going to do this I would probably bypass the hot air balloons altogether because the highest rope tied to the satalite would only be able to grab the cargo once every orbit. With wind shifts in the lower-upper atmosphere it might be very hard estimate where the rope could catch it's load. I suggest that you just take a jet carrying the cargo to the rope attached to the satelite and hand off like those refueling jets do to sustain long flights. then you have to make sure your satelite can thrust to gain speed to counteract its new weight. That way it won't come crashing back down. Then you just lift away.