Any kind of statement, confirmation, or receipt should be printable. Some places still require printed documentation of expenses for reimbursement (even if it's only to turn around and scan it; you can argue it's silly but that doesn't mean it doesn't happen). A paper print is also an easy, cheap, immutable and durable one-off backup for anything you're looking at on your screen.

Yes. Also any press release, privacy policy, terms, ideally your contact page and user account screens. Stuff that might come up in a dispute or might be necessary in PDF form in a handover document. And any documentation for a physical, offline device or appliance.

From this I think it becomes obvious that the best strategy is to make the site as a whole at least not a total train wreck when printed, and then to focus effort a little more on the specific screens.

People who print are quite tolerant of small issues, I find. They are used to much worse.