Advanced Linear Algebra
This class is the gateway from the undergraduate math classes to the graduate ones. You touch on applications to linear algebra to a lot of areas in both pure and applied math. The difficulty obviously depends on your background (It is a lot easier if you have taken some sort of course in abstract algebra), but this was probably the hardest math class I've taken at Duke.
Advanced linear is for the serious math student. It is a hefty class but man is it fun. It is much more difficult than 221 but is good for grad school apps and great if you enjoyed linear.