Extremely hard and stimulating (undergraduate) real analysis $problems$

You could try the book "Selected problems in real analysis" by Makarov and 3 co-authors: http://www.amazon.com/Selected-Problems-Translations-Mathematical-Monographs/dp/0821845594

This is a very beautiful book, and some of the problems are quite hard.

I recommend G. H. Hardy, "A Course of Pure Mathematics", a classic with many challenging problems. http://ebookee.org/A-Course-of-Pure-Mathematics-Centenary-edition_731929.html

Here are two suggestions:

-- A link to Vaughan Jones's RA course at Berkeley. In the introductory remarks he acknowledges the difficulty of the HW problems (links on the page)

https://math.berkeley.edu/~vfr/MATH10411/index.html

-- Pugh's "Real Mathematical Analysis," a great book in its own right, has over 500 problems, with many from Berkeley qualifiers.

http://www.amazon.com/Mathematical-Analysis-Undergraduate-Texts-Mathematics/dp/0387952977