I do a couple questions regularly with my BFF. We used to do it over dinner every night when we were in college and actually saw each other. I honestly think they're things everyone should ask everyone they care about on a daily basis, and they're also good to just get to know people better. You can ask follow-up questions or not depending on how deep you want to get.
What's something you learned today/this week?
What's something that made you sad today?
What are you looking forward to the most right now?
What's something that made you laugh this week?
I asked my now wife what her favorite childhood book was. The book she picked as her favorite is one I remember my mom and grandma both reading to me when I was a kid. I used it to propose to her only a couple weeks after meeting her. We've been married five years now and that was one question that really helped me get to know her on a deeper level. That book was "Love you forever" by Robert Munsch. The most memorable part went like this;
"I'll love you forever,
I'll like you for always
As long as I'm living
My baby you'll be."