One of my favourite conversation-deepening questions is "What's something you like about yourself that other people usually notice, and what's something you like about yourself that other people rarely notice?"
Plenty more. I've written maybe 200-300 questions over the years (and asked thousands more).
I'm thinking to publish more of them in a neat and useful format soon for easy access and use. Should be ready in the next 2-4 weeks.
If you'd like to be notified when I do, my email list on the site might be the best bet (or just following me on twitter @ankitshah). Not sure I'll remember to come back to this thread to post.
You might enjoy the Authentic Relating Games collected by Sara Ness at https://www.authrev.org/. I've played a few and they have been deeply connecting.
I tried working through what my answer would be for them, and it felt like a good jumping off point for forced introspection. The one about a lie you tell often was particularly good lol.