Mathematical literature to lose yourself in H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics:

...I have tried to say to students of mathematics that they should read the classics and beware of secondary sources

In trying to learn more, I feel that I have accumulated a bookshelf full of secondary sources which I have left largely unread. So I would like to take heed of Edwards's advice and read some classics for the wonderment of it and for the respect I would gain for mathematicians of the past. I would like to exercise creativity over rigour, at least for a little while.
But this begs the question: what are the classics? And this is what I hoped to ask about.
Which primary sources do you feel are suitable for self-study? Naturally this is a broad question, but people must have their favourites, and I am hoping for some recommendations on the basis of these personal preferences.
 A: For introductory Number Theory, you could go with Gauss, Disquisitiones Arithmeticae. Don't worry, you don't have to read Latin, it is available in English and other living languages. 
A: Three books: Euler's Introduction to the Analysis of the Infinite and Foundations of the Differential Calculus both translated by JD Blanton and published by Springer, also the very informative Analysis by its History by Hairer and Wanner. There are always the original papers by the biggies which are more often than not very interesting, illuminating and convey a sense of a firsthand encounter with the author(s).
A: Euclid's "The Elements".  Greek.  Old.  It doesn't get much more classic than that.  As well as his other writings.  http://en.wikipedia.org/wiki/Euclid#Other_works
Also, the Principia Mathematica by Newton. As well as his other writings.  http://en.wikipedia.org/wiki/Isaac_Newton#Mathematics 
Archimedes probably deserves a mention as well.  You know, pi and circles and all that.  http://en.wikipedia.org/wiki/Archimedes#Writings
Also I've seen Gödel, Escher, Bach mentioned on this page several times.  Perhaps not canonically 'classic'.  But really an excellent book.  Changed the way I've thought about life, science, philosophy, CS, and many other things.  It's also probably a good gateway into Gödel's incompleteness theorem.
A: There are several Source Books that have made nice selections for you to pick from, e.g., Smith, Struik, Fauvel and Gray, Stedall. But for an extended read, you can do nothing wrong by immersing yourself in Gauss' Disquisitiones.
A: I'm not a mathematician and I'm advocating a secondary source here, but I'm very convinced that this book fully measures up to the greatness of the original classics: My personal favorite is Folland (1999).
I had been a total newbie to measure theory, topology, and functional analysis. It took me about 1.5 years to carefully study the first seven chapters (the book is extremely dense—it's like the Nutella of math textbooks), and it gave me a very deep working knowledge of real analysis. I owe more than 95% of everything I know about advanced mathematics to this fantastic textbook.
In addition, each chapter ends with a refined overview about the most important and distinguished classic articles and books the chapter builds on: It is worth having a look at these notes even if you are more interested in the original sources.
A: I should say that as far as recreational maths is concerned and you have tagged this question as such, going through this book will be a very satisfying experience: Winning Ways for Your Mathematical Plays (all volumes). Games were not taken seriously by mathematicians as you will recall Euler himself dismissing the graph theory field as not serious maths even after having solved Seven Bridges of Königsberg. This book tries to go that extra mile.
A: If you want to learn algebraic geometry, a classical paper is Jean-Pierre Serre's FAC. See here for the French original, and here for the Englisch translation. See here for some advertisement by Georges Elencwajg.
A: Not exactly what you were looking for (because these are "secondary sources"), but maybe interesting nevertheless:


*

*Mathematics and Its History by Stillwell walks you through the history of mathematics showing original problems in modern notation with many good exercises at an undergraduate level and with lots of pointers to the original sources.

*Euler - The Master of Us All by Dunham provides a guided tour through the main ideas of many of Euler's original papers.   (And fittingly starts with a Laplace quote: "Read Euler! Read Euler!")
A: Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter.
Although not strictly speaking a purely mathematical book, surely I would put it among the classics.
A: Categories for the Working Mathematician by Saunders Mac Lane, is not technically primary, since it is not the "founding papers", but it is excellent, broad and exquisitely beautiful. And Saunders Mac Lane is one of the founders of category theory.
