Tethered Satellite.
This applet shows a satellite at 60,000 km altitude tethered to the equator of the planet. The satellite orbit is geosynchronous. The satellite can be released by clicking the Release/Shatter option.
Since satellites tethered in this way are geosynchronous, a whole set of satellites at different altitudes tethered to one point on the planet equator would form a tower radiating out from that point. However, those in low orbits ( < 42,000 km radius ) would fall to earth unless they were held up in some way. The one obvious way to do this is to connect low altitude satellites to higher orbit satellites, creating a chain of linked satellites. The next applet explores this possibility.