It very much depends on the situation and what kind of intimate detail you are trying to figure out. Is it a job interview? wanting to have sex with them? trying to get a friend better? trying to get to know a possible partner better?
However there are some things that can tell you a lot about a person if they choose to share information.
1) What's your biggest fear?
2) what is your first childhood memory?
3) what do you want to be when you grow up? (basically a cooler way of asking what's their dream job)
4) what are your favorite movies \ books \ music albums \ artist?
5) what's your relationship with your parents like?
6) have you ever had your heart broken?
7) what was the most scary things that ever happened to you?
8) have you ever been in a long term relationship? if it end then why?
9) what's the most crazy \ wild thing to ever did?
10) what was your first sexual experience like?