Is an internet connection required after the initial setup of the Xbox One?

An internet connection is not required per se. Of course, for activities that need internet, such as multiplayer, it is required. Some games require an online sign-in (for example Destiny), so you cannot play the singleplayer either.

However, for the majority of games, it doesn't matter that you are offline, in regard of playability. There are no updates, for both the game and the Xbox, but that if that is not a dealbreaker, then you are good to go. Achievements etc can still be obtained, but you'll only be awarded them when you come online. So if you did achieve something, you won't get the pop-up and points until you come online.

you surely need an internet connection, mostly all the time you try to play a game, an update its required, and you have to be online, or cant play the game, i update a game and play it one day, the next it already asking for a damn update, and its a pain, because i have a slow internet connection, i hate xbox one, its terrible how many times it asks you for an update, cant even play.