A building that is actually a whole city; trust the Japanese to be ambitious enough to try it:
"The X-Seed 4000 proposed 4000metre (13,123 foot) height, 6 kilometre wide sea-base, and 800 floor capacity could accommodate five hundred thousand to one million inhabitants. Designed by Taisei Construction Corporation, as an “intelligent building” the super-futuristically-named X-Seed 4000 would maintain light, temperature, and air pressure in response to changing external weather conditions. Because it has already been fully designed using materials available today, the structure could, in principle, be built, although it would likely cost several hundred billion dollars, if not more. Because the structure would weigh so much, it could only be built on the sea if present-day construction materials were used.
Unlike conventional skyscrapers, the X-Seed 4000 would be required to actively protect its occupants from considerable air pressure gradations and weather fluctuations along its massive elevation. Its design calls for the use of solar power to maintain internal environmental conditions. Some estimate that the cost to construct the X-Seed 4000 structure may be somewhere between US$300-900 billion, in current dollars."
Thanks to the Eyebeam team for the heads-up and words.