World's Tallest Buildings <=> Highest Radio Masts
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When a structure is measured for an official height, it includes every part of the structure - including the antenna(s) and support mast(s) that is(are) almost always included at the very top. Take the current height record holder, the Burj Khalifa, in Dubai, for instance. The overall height of the structure including the antenna is 828 meters; however, the antenna is 207 meters tall, so the building itself is only 621 meters. As can be seen from the chart below, only two pure antennas (including support mast) are included in the world's tallest structures. Antennas are used for television, radio, data, and even optical transmission.

If you read Arthur C. Clarke's article in the October 1954 edition of Wireless World where he conceived of a geosynchronous satellite system to broadcast television signals rather than using a series of terrestrial towers, it is apparent why even using antennas at heights like the one on the Burj Khalifa would not even come close to providing coverage needed for the entire earth. The following table gives the line-of-sight*, including earth curvature, range for each antenna (assuming maximum specified height). With an earth circumference of 40,000 km, you would need nearly 700 towers just to provide continuous coverage along the equator. A few thousand would be required for global coverage. The logistics for construction, maintenance, and operation of such a network would be daunting... if not impossible. Imagine the aerial navigation that such an array would pose!