I dont believe the movies/shows have really touched on this much (which is a shame), but my understanding is that the biggest difficulty in constructing the Death Star is to get the superlaser working right.

The Catalyst Rogue One prequel novel goes into this a lot, and you hear some brief mentions of it in Rogue One (Krennic tells Galen that the work has stalled in his absence, Galen's contribution to the Death Star is his work with kyber crystals which power the superlaser).