If it is the very end of the interview, a classic question is "what are our next steps?" You want to know whether to expect another interview or if they're going straight to a decision, and also roughly when. This is the note on which the conversation ends, so save it for last.

Before that, don't worry too much about questions that "show you care" and "are eager". As an interviewer, I make note of many things, but what I was asked is rarely specifically important. Ask what you want to know. If you really have no questions, say "I had a number of things I wanted to find out about, but you've covered all of them, thank you!" and then ask what the next steps are if you feel the interview is wrapping up. (Don't ask it if 10 minutes into a one hour interview you are asked "but before I go into detail on that, is there anything you wanted to ask?")