If you are looking for some challenging calculus problems, I would suggest looking at the HMMT (Harvard-MIT Mathematics Tournament) problems archive. Take a look at the calculus test for the years 1998-2011. I don't think these problems are the routine exercises that you have grown weary of in Thomas' Calculus.
If you want another book that may suit your taste, I recommend "Introduction to Calculus and Analysis" by Richard Courant and Fritz John. I think the problems are a bit harder than those in Thomas' text. But more importantly, Courant and John give a lot of motivation for calculus through discussing its relevance to the physical world (there is even discussion of Fourier Series in the study of a vibrating drum).
Courant and John also introduce some concepts of modern analysis (calculus done more rigorously) such as monotone sequences, a precise discussion of a limit, etc towards the end of the book, if that interests you.
Last but not least, I see that you mentioned the Putnam. I think a great book for doing calculus problems that may prepare you for the Putnam is Titu Andreescu's book- "Problems in Real Analysis: Advanced Calculus on the Real Line".
Andreescu's texts are known to be useful in preparing students for mathematics competitions. In this textbook, Andreescu exposes you to a lot of techniques (especially in dealing with inequalities) that may help you with the Putnam. You may also learn some analysis along the way, but since he is dealing only with the real line, he doesn't "weigh you down" with various topological concepts that may or may not interest you. If you have mastered the concepts of calculus and are looking for a challenge, this book may be for you.
Good luck!
Donald J. Newman’s A Problem Seminar is a classic and a delight, and you will certainly benefit from it. It is not a textbook, because it does not teach advanced theorems - it is specifically intended to get your mind aligned to problem-solving.
It has a hundred problems (two or three lines long at the most), then a section with a brief hint as to how to approach each one; and finally the main body of the book gives the answer to each.
The main thing about problem-solving as a specific skill is that it is a question of recognition rather than heavy lifting. The tools in Newman’s book are all within the reach of a high school mathematician. The reason the book is a challenging exercise even for university students is that knowing which tool to use, the key to all advanced mathematics, is a subtle and elusive skill, hard to learn, you might say impossible to teach. But it is a delight.
The book is rather expensive, new, and you might have to sell your grandmother to buy it. But you may be able to find a second-hand copy more reasonably: it came out in 1983.
Best Answer
If you are looking a book which has plenty of exercises on basic calculus, you could try B.P. Demidovich: Problems in mathematical analysis (originally Demidovich: Zbornik zadac i uprazneni po matematicheskomu analizu, there are translations to several languages). Even Russian edition might be good for you - for computing problems, like calculating a limit, a derivative, an integral you do not need understand the text of exercise. You can probably find it online, e.g. try searching for: demidovich zbornik djvu or Демидович Сборник задач djvu or demidovich problems djvu and similar searches (you can try pdf instead of djvu).
I very much like the selection of exercises in this book.
Kaczor, W. J.; Nowak, M. T. Problems in mathematical analysis. I. Real numbers, sequences and series. Google Books MR1751334
This book has two more volumes (II. Continuity and differentiation and III. Integration). There exists English, French and Polish edition.
Many interesting problems on real analysis are in
Rădulescu, Teodora-Liliana T.; Rădulescu, Vicenţiu D.; Andreescu, Titu. Problems in real analysis. Advanced calculus on the real axis. Google Books, MR2514007.
Although I am not sure whether this is at the level you're aiming for. (But you asked about real analysis in your question, too.)