Non-multiplayer games, clock, camera, contacts, phone, text message, file explorer, keyboard, launcher, notes, document viewer/editor, image viewer, audio recorder...
Most of the apps on my phone do not need internet access.
That said, I'd love to have a new "Internet access" permission for apps, so users had the choice. Perhaps even separate "Allow iCloud" and "Allow Internet" but that's probably too granular for Apple's taste.
I have no idea if this is what already happens, but I feel like it might be. (Why would each app have all these network connections when the system could just manage it instead?)