Extremely hard and stimulating (undergraduate) real analysis $problems$ To put it simply: I have seen many problem books in real analysis (also on this website), but the exercises they propose seem quite standardized. 

What are problem books that propose really challenging and stimulating
  problems (as opposed to standardized exercises)?

 A: 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.
A: 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
A: 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
A: Here is a list of undergrad level analysis problems intended to prepare you (in part) for an analysis graduate qualifying exam.
You can find lots of undergrad level problems in qualifying exams from many different departments. For instance, you can get Wisconsin's here.
A: Try "Problems in mathematical analysis" by Kaczor Nowak, AMS. This book has three volumes and you need them all.
For metric spaces I prefer http://www.amazon.in/Topology-Metric-Spaces-Second-S-Kumaresan/dp/8184870582 this book and http://www.springer.com/in/book/9781846283697
lastly for topology I prefer http://www.amazon.in/Topology-Munkres/dp/9332549532/ref=sr_1_1?ie=UTF8&qid=1495724772&sr=8-1&keywords=munkres+topology , 
https://www.amazon.com/Topology-James-Dugundji/dp/B002OOA0VO 
and http://www.amazon.in/INTRODUCTION-TOPOLOGY-MODERN-ANALYSIS-Simmons/dp/0070597847/ref=sr_1_1?s=books&ie=UTF8&qid=1495724849&sr=1-1&keywords=simmons+topology
lastly as an undergraduate student I should strongly recommend baby Rudin book(principles of mathematical analysis),this book's problems are really good! and everyone should try them.
