I am going to copy and paste my answer from another question on this site, because I think one would be hard pressed to beat it in terms of the number of suggestions it covers, and the general quality with which it presents these suggestions:
You might be interested in the expansive answers that were generated on math.stackexchange by the questions Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…” and Book series like AMS' Student Mathematical Library?.
In order to make this answer complete in its own right, and in order to reduce the number of times one has to depress a mouse button, I will summarize the suggestions from those threads here, giving credit to those who originally provided them.
“…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”
Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…” was asked due to inspiration from from Charles Radin's Miles of Tiles, which has the following description:
Theme: "In this book, we try to display the value (and joy!) of starting from a mathematically amorphous problem and combining ideas from diverse sources to produce new and significant mathematics--mathematics unforeseen from the motivating problem ... "
Style: The common thread throughout this book is <insert topic here>...the presentation uses many different areas of mathematics and physics to analyze features of <insert topic here>...[as] understanding <insert topic here> requires an unusual variety of specialties...this interdisciplinary approach also leads to new mathematics seemingly unrelated to <insert topic here>...
Readership: Advanced undergraduates, graduate students, and research mathematicians.
mweiss further suggested Soifer's How does One Cut a Triangle?:
You may enjoy Alexander Soifer's book How Does One Cut a Triangle?. From my review of this on Math Reviews (MR#2548775):
Indeed the entire work is a sequence of problems posed and solved, with each new solution yielding, through generalization and specialization, new questions. One of the most noteworthy features of the text is its “just-in-time” approach to introducing new ideas: tools from linear algebra (linear independence and eigenvalues), Diophantine and algebraic equations, calculus (the intermediate value theorem), combinatorics (the pigeonhole principle), and affine geometry are brought in with a minimum of fuss precisely when they are most useful.
Conifold further provided an absolute wealth of suggestions:
1) Books with light prerequisites
Stories of Maxima and Minima by Tikhomirov, a guided tour of extremal problems starting with Dido and the founding of Carthage all the way to convex programming with geometry, optics and mechanics visited along the way. While the author aims the book at "high school students" he means Russian ones perhaps.
Indra's Pearls: The Vision of Felix Klein has Mumford (that one) for one of the authors, and a wikipedia article devoted to it, saves me the effort.
Gödel, Escher, Bach by Hofstadter is a book with almost cult following, also has a wikipedia article. Very roughly, looks into how recursion and self-reference lead to expressing meaning in formal systems, music and art. Goes in depth into Gödel's incompleteness and mathematical themes of Escher and Bach, while staying a literary marvel that won a Pulitzer prize. According to Martin Gardner, "a book of such depth, clarity, range, wit, beauty and originality that it is recognized at once as a major literary event".
Fibonacci Numbers by Vorobiev studies the title subject by introducing modular arithmetic, recurrence relations and continued fractions, then discusses their role in approximating irrationals by fractions, Fibonacci enumeration system for integers and its application to winning a Chinese game, their appearence in geometry alongside the golden ratio, and in the theory of search.
Mathematical Gift I-III by Ueno, Shiga and Morita is a well designed intuitive transition into graduate notions of geometry and topology, with highlights including Poincare-Hopf and Gauss-Bonet theorems, theories of dimension and volume (with Banach-Tarsky paradox explained), Poncelet closure theorem in projective geometry, Whitney embedding theorem, and Dehn's solution to the third Hilbert problem.
Felix Klein and Sophus Lie by Yaglom is an inspired story of how a mathematical theory is born, the theory of symmetry. The content is much broader than the title, related ideas of Galois, Poncelet, Hamilton, Grassmann, Cayley, Peirce, Clifford are thoroughly explored as well. Most insightful historical account of 19th century geometry and algebra.
Knot Book by Colin Adams is a gem that takes one from knotting and braiding rope to topological invariants, Seifert surfaces, 3-manifolds by surgery and applications in biology, chemistry and physics.
Excursions into Mathematics by Beck, Bleicher and Crowe is a collection of 6 mini-books under one cover. My favorite ones are on perfect numbers, the ancient topic that launched much of modern number theory (which still can't answer some basic questions about them), and on exotic geometries. You may like that one because it comes close to "laying down the axioms and playing with them" from your other question, albeit in geometry rather than algebra. From Euclid's postulates to Hilbert's axioms, what happens if some of them are modified, on to Latin squares, arithmetic of finite fields, lines and circles in finite projective spaces, and geometries they create.
Fearless symmetry by Ash and Gross is not a text for liberal arts majors despite the title. It sets out to outline a proof of the Last Fermat Theorem to non-experts with all the jazz of quadratic reciprocity, modular forms, algebraic integers, Galois group of $\mathbb{Q}$ and its representations on elliptic curves, traces of Frobenius elements, etc.
Zermelo's Axiom of Choice by Moore. AC with its controversies and history up to Gödel and Cohen, and equivalents and consequences in algebra, topology and analysis.
Proofs and Confirmations by Bressoud follows your wishes very closely. It is a thrilling story of proving a conjecture about the total number of alternating sign matrices that draws on insights about partitions, symmetric functions, hypergeometric series, lattice paths and statistical mechanics.
2) Advanced Books
Ramanujan by Hardy is not a biography but a look at Ramanujan's enigmatic mathematical legacy by the man who knew him best. Hardy explains and ties together Ramanujan's 'magic' insights into primes, partitions, hypergeometric series, zeta function, elliptic and modular forms. Understanding the genesis of analytic number theory is a side bonus.
Exploring the Number Jungle by Burger. The theme is approximating irrationals by fractions with relatively small denominators, a.k.a. Diophantine approximation. But that doesn't stop Riemann surfaces, elliptic curves, Pythagorean triples, quadratic forms and $p$-adic numbers from showing up. It is unusually written: there are descriptions, questions, theorems, exercises, hints, but no proofs. On principle.
Radical Approach to Real Analysis also by Bressoud is a very unconventional exposition of the subject that starts with the crisis in mathematics posed by the discovery of Fourier series and develops ideas in a very versatile manner, highlighting perspectives lost in modern texts.
Mathematical Coloring Book by Soifer, who also wrote How Does One Cut a Triangle. Coloring everything here: polygons, graphs, plane, space, integers, arithmetic progressions, but it all ties to the chromatic number of the plane. Which depends on the axiom of choice and existence of inaccessible cardinals (not kidding!).
Glimpses of Soliton Theory by Kasman is a rare book on the subject that doesn't just throw cumbersome computations and transformations at the reader. Intuition for non-linear PDE-s is built up through examples and history, and then supplemented with ideas about elliptic curves, isospectrality, wedge products, pseudo-differential operators and the Grassmann cone.
Tour Through Mathematical Logic by Wolf is a historically driven exposition of advanced modern logic including Gödel's incompleteness and constructible hierarchy, model theory, Cohen's forcing, Robinson's non-standard and Bishop's constructive analyses, large cardinals, determinacy and the Woodin program.
Mathematical Methods of Classical Mechanics, Arnold's classic, is about mechanics obviously. And about differential forms, Poisson structures, symplectic manifolds, geodesic flows, Legendre transforms and singularities, to name a few. According to a MathSciNet reviewer a unique element in the intersection of "the most influential books of the second half of this century, the most frequently quoted books, books that have the highest probability of surviving into the 21st century, books that are very useful in teaching, books characterized by a very strong personal style, books that provide a delightful reading experience."
Free electronic versions may be available here or here.
Some more suggestions were provided by user89:
Counting on Frameworks ...fits the style of my original request. The author builds up a mathematically amorphous problem ("what is a rigid structure?"), progressively developing a theory (along with the participation of the reader through exercises) to describe "rigidity".
Combinatorics through guided discovery, by Kenneth Bogart. It seems to be quite fun! You learn combinatorics by going through exercises, rather than by being told what it is -- in other words, you make up the subject yourself, in a guided way. The book is completely free, and available online.
Formal Methods of Software Design, available online for free..., is an excellent introduction to boolean logic, and general theory building -- it has been a fantastic base for learning other math. As a cool side effect (or was that the original goal?), you'll learn how to derive computer programs (yes, computer programs), from boolean specifications: very much like how you would derive any other proof! Fantastic.
Learning how to Learn ...(offered by the UC San Diego, on Coursera), which is actually, hands down, the most honest and effective course (good MOOCs are rare these days) on the subject that I have ever come across. Generally useful ideas to keep in mind while learning new mathematics!
Book series like AMS' Student Mathematical Library
This reference request was created because:
I had the joy of discovering AMS' Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect stepping stone for learning!
I am familar with some Springer book series (Undergraduate/Graduate Texts in Mathematics), but I think those have a much more of a textbook nature in general.
What are some great book series that fit the style of Student Mathematical Library?
See this question for inspiration as to what the answers should look like.
Conifold once again delivers an excellent answer:
Generally what I think makes such series so good is that the format forces the authors to explain non-trivial and often non-elementary mathematics in accessible and inspiring way. The concentration of mathematically "clever" and "cool" both fascinates and challenges. They also often expose parts and perspectives of mathematics that are largely missing in standard texts and approaches. Of course, not every book in a series is equally good, so I will list some that I find particularly outstanding. But I haven't read all of them, so it doesn't mean that the rest are sub par, and my assessment of level only applies on average.
Not a series per se but similar in spirit and close to the upper undergraduate level of AMS Student Mathematical Library (and cheap) are some (most are just texts) of the Dover Books on Mathematics: Riemann's Zeta Function; Three Pearls of Number Theory; Geometry and Light; Counterexamples in Topology; Regular polytopes; Beauty of Geometry; Asymptotic methods in Analysis; Satan, Cantor and Infinity; Hyperbolic Functions.
Some of Cambridge University Press' London Mathematical Society Student Texts are more than typical texts, and they are at the right level too: Prime Number Theorem, Undergraduate Algebraic Geometry, Elliptic Functions, Young Tableaux. Also good but very short are their Outlooks, and Canadian Mathematical Society's Treatises in Mathematics series.
MAA and Cambridge University Press support Dolciani Mathematical Expositions, which is freshman/sophomore level: Charming Proofs; Diophantus and Diophantine equations; Logic as Algebra. MAA's Classroom Resource Materials also has some entries at this level: Paradoxes and Sophisms in Calculus; Counterexamples in Calculus; Explorations in Complex Analysis; Which Numbers are Real?; Real Infinite Series.
The ones below, especially Mir's, are generally less advanced, high school/freshman level. Still, I grew up reading such booklets, and learned from them more than from most formal studying, they also guided my interests later and helped select topics which I wanted to pursue in depth.
AMS's Mathematical World: A Mathematical Gift, I, II, III; Mathematical Ciphers: From Caesar to RSA; Kvant Selecta (collections of best articles from Russian math journal for advanced high school kids); Stories about Maxima and Minima.
MAA's New Mathematical Library: Game Theory and Strategy; Geometry of Numbers; Numbers: rational and irrational; Ingenuity in Mathematics; Geometric transformations; Uses of Infinity.
Mir's Little Mathematics Library: Proof in Geometry; Solving Equations in Integers; Inequalities; Areas and Logarithms; Remarkable Curves.
