Paper cups contain either wax or plastic to avoid liquid from soaking through. They must always be thrown in the trash. However, cardboard sleeves can be recycled.
Local coffee shops offer discounts if you bring your own reusable cup (shown below):
Coffee Bar: $0.30 off all day
4-6pm happy hour: $2 any size drink in your own cup
Wild Cherries: none
Dark Horse: $0.25
Tuff Beanz: Charges for a smaller size than what fits in cup.
Plain coffee is just $1.