I don`t know if this is the most awkward, but is one i remember right now:
Many years ago, when i was in a book sale with my family. A random girl (i think she was between 12 or 13 years old), came to me and the first thing she says is this:
-"do you want to be my friend?".
I was confused and surprised, i said "okay", because i guess i was being polite.
She was following me around while i checked some books.
After a little silence, she asked me my messenger account (it was older times)
A long awkward silence later, a child dragged her out.