Like Asimov explicated in some of his later Foundation books, a planetwide city like Trantor or Corusant would need to incorporate some "planetary functions" into its infrastructure. There'd have to be closed-cycle life support systems for recycling not just air and water but also vital basic minerals such as iron and phosphorus; massive heat transfer and dissipation arrangements; even mechanical arrangements to accommodate continental drift (or, if the planet has no continental plates, cooling shrinkage). Some of this would have to be installed already when the cities began to cover 10% or so of the surface, and that's assuming the cities were eco-friendly green wonders that didn't ruin the planet at 1% coverage already.
However, none of that is a reason not to do it. Having your city spread across the surface of a planet means having it situated atop a vast trove of raw materials, and a fairly effective heat source or heat buffer, at least in the short term of mere tens of millennia. It's fairly irrelevant whether you build up or down, as the city will soon start spanning several vertical kilometers anyway, and the life support requirements aboveclouds and underground are not that dissimilar. But of course building up will have its advantages in that the buildings will soon be rid of the troublesome atmosphere, and can more efficiently transfer materials and information with the outer space - or with each other, via aerial means.
And soon enough, such construction will require advances in materials tech that allow you to build beanstalks for efficient surface-to-orbit access. It might even be that the surface layer of cities may briefly shrink as industries and special functions crawl up the beanstalks, giving the planet an equatorial mohawk (and probably causing all-new climatological problems that call for climatotechnological solutions).
A few arcologies here or there on Iowan cornfields won't yet call for such engineering, though. They, too, will probably initially siphon other sorts of habitation off the surface, reducing urban sprawl. Although whether that will be in favor of wilderness and recreational spaces or intensive agriculture will depend on how the arcologies produce their food.
Timo Saloniemi