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"[Science fiction] is the one literary medium left in which we have a free hand. We can do any damn thing we please."
- Alfred Bester

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...

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.

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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