I'd imagine quite a few.
Anyone wanting to bring an MMO to the console, for example. That's the most obvious case.
Anyone wanting to have a single player game experience that can drastically change at any momemt is a non-obvious case.
Anyone wanting a game that is server/client based such that game balance and AI bug fixes, etc. could be changed/fixed at any moment without requiring the user to download a patch because all changes are server side only is another non-obvious case.
In that case, an RPG could introduce daily/weekly/monthly quests on the server without requiring the client (player) to download anything as long as no art assets were added or changed. Difficulty could then also be dynamically adjusted at the server for single players or all players based on how all connected players were doing.
Imagine a Police detective type of game where you have to research things in game online. Only that the stuff you have to research changes constantly each time you play. In addition when research the data for the case in game, you're actually accessing real internet information to do it. So instead of being limited by how much data can be fit onto the optical disk and hard drive, you can include any and all information available on the internet. Say you have to research some historical figure to solve a crime. Instead of the tiny bit of information games typically include for such things you can access a computer terminal in game which then allows you to research anything on the actual internet about said historical personality.
Your game world could then theoretically have more content than could fit onto 1, 2, 3, or more BluRay disks as long as the major art assets reside on the console. Although this would require a revolution in game design, as the majority of the size of a game install is art assets, sound, and video.
The possibilities are endless for game enhancing features that require a player to always be connected.
And, of course, any publisher that is worried that said console might get hacked and the game thusly get pirated. Or a developer making an multiplatform game that wants to make the PC version as difficult to pirate as possible would necessitate the console version also being that way (IE - client/server based game with as much stuff done and hosted on the server as possible).
Regards,
SB