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"The answer to the problem of information overload on the Net is reputations… engineer a system called a reputation server."
- Neal Stephenson

Trantor  
  A city that covers the entire surface of the planet.  

The logical extension of urban sprawl.

Trantor... could scarcely avoid being the densest and richest clot of humanity the Race had ever seen.

Its urbanization, progressing steadily, had finally reached the ultimate. All the land surface of Trantor, 75,000,000 square miles in extent, was a single city. Its population, at its height, was well in excess of forty billions...

Technovelgy from Foundation, by Isaac Asimov.
Published by Doubleday in 1951
Additional resources -

Science fiction writers have created (mostly) dystopian stories about enormous cities; for example, The Sprawl or the Boston-Atlanta Metropolitan Axis (BAMA) from William Gibson's work and Mega-City One from the Judge Dredd comic.

Fans of sf great Clifford Simak may recall the metal calculator planet; it was a regular planet covered with machinery to a depth of twenty miles.

Harl Vincent gave a preview of this way of building in his 1929 story The War of the Planets:

Washington was one of the few cities in the world that still retained the old arrangement of wide streets, spacious detached dwellings, and pedestrian traffic. Of course, there were landing stages on all buildings for the aeros, but there was none of the closely massed, continuous building construction with roofed-over multiple moving ways and artificial temperature control and ventilation encountered almost everywhere else in the world. Here one could look at the stars without taking a long elevator journey to the roof-tops of a completely covered city.

Compare to planet city from The Message from Space (1930) by David M. Speaker.

Thanks to an anonymous reader for reminding me to add this idea.

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Additional resources:
  More Ideas and Technology from Foundation
  More Ideas and Technology by Isaac Asimov
  Tech news articles related to Foundation
  Tech news articles related to works by Isaac Asimov

Trantor-related news articles:
  - China's Megacity 'Turn The Pearl River Delta Into One'

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