In many very popular sci-fi anime stories, there are "space elevators", or, in other words, a giant tower which allows to transport people and/or objects from the surface of Earth to space. Some of these stories where they appear are Gundam (Gundam Reconguista in G (2014) , Mobile Suit Gundam 00 (2007) , Turn a Gundam (1999)), Eureka Seven (2005) and Bubblegum Crisis Tokyo 2040 (1998) and the earliest I know it's Super Dimensional Century Orguss (1983). Now, I know this idea comes from science, since a giant tower which could reach space from the surface of Earth was theorized by scientists (though apparently a material which could have the properties to build it doesnt exist) but I wonder which was the first sci-fi story to give credit to this concept by incorporating it into a fictional story.

Which was the first story to feature space elevators?

The Fountains of Paradise (1979)

Jerome Pearson, President of STAR, Inc., conceived the idea of the space elevator in 1969 at the NASA Ames Research Center, and perfected the concept in the early 1970s, when he was at the Air Force Research Laboratory in Ohio. He published his ideas in an international journal that first brought the idea to the attention of the entire world of spaceflight researchers. Sir Arthur Clarke, living in Sri Lanka, consulted with Pearson in the late 1970s in writing his novel, "The Fountains of Paradise," which brought Pearson’s idea of the space elevator to an even larger audience. Sir Arthur included in the book an appendix that credited Pearson.
_{- Space Elevator History - Star Tech Inc}

So it seems like this was the first one.

Arthur C. Clarke's The Fountains of Paradise and Charles Sheffield's The Web Between the Worlds (both published in 1979) are generally considered to be the works that introduced space elevators to the science fiction community at large.

Charlie and the Great Glass Elevator by Roald Dahl (1972) has an elevator that goes into space, but presented as an absurdity with no scientific explanation.

Both Wikipedia and TVTropes have extensive lists of fictional space elevators, though they're not chronological.

In the Jan 12, 2019 issue of New Scientist magazine, an article by Kelly Oakes (on plans for real-word space elevators) mentions 2 sources for the history of the idea:

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  (probably non-fiction only) Konstantin Tsiolkovsky¹ in 1895 "imagined a 'celestial castle' orbiting the earth at 36000 km (geosynchronous), attached to the Eiffel Tower in Paris by a long spindle." This was ~70 years before the first geosynchronous satellites were launched.

  
  
  

  http://pages.erau.edu/~andrewsa/history.html briefly summarizes the "celestial castle" idea. Tsiolkovsky did apparently write science fiction as well, but I haven't been able to find a reference to a story involving the idea.

  
  
  

  1: yes, the same guy the rocket equation is named for

  
  



• Arthur C. Clarke's 1979 novel The Fountains of Paradise "popularized an elevator proper".

Clarke's story is definitely the most widely cited early fictional version in discussions I've seen of space elevators. But perhaps Clarke didn't invent the entire concept from nothing. (That's not the question, and Clarke may still have had to figure out plenty of practical details.)

The article says that we now know that a real elevator needs to stretch beyond geosynchronous / geostationary orbit altitude to bring the centre of mass up beyond that point. (Otherwise the cable falls under its own weight.) Apparently Clarke pictured the top being at around geosync altitude, too.

A large counterweight just beyond that altitude should work, but the article suggests having the cable extend to something like 100 000 km to counter the weight of loads traveling up/down the cable. That would give you a nice escape trajectory for leaving Earth orbit.

Further material:

Jerome Pearson wrote (in 1997) a paper on the history of the ideas / invention of Space Elevators. He doesn't consider Tsiolkovsky's thought experiment / concept to be detailed or practical enough to call it "inventing" the space elevator.

He credits the invention of the space elevator as an actual engineering problem to Yuri Artsutanov (1960) and independently to himself (1975), as Artsutanov didn't publish a paper so it wasn't known in the west.

Vathek (1786)

One of the earliest sci-fi stories has this plot element. In the novel Vathek by William Beckford, the titular character builds something like a space elevator to do astronomy:

To better study astronomy, he builds an observation tower with 11,000 steps.