Books you would like to read (if somebody would just write them…) I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different approach.
(I do hope this is not inappropriate for MO.)
Let me start with some books I would like to read (again with self-explanatory titles).

*

*The Weil conjectures for dummies


*2-categories for the working mathematician


*Representations of groups: Linear and permutation representations made side by side


*The Burnside ring


*A functor of points approach to algebraic geometry


*Profinite groups: An approach through examples
Any other suggestions ?
 A: AD${}^+$ by Hugh Woodin.
A: Inter-Universal Geometry for dummies.
A: Nonlinear Differential Galois Theory
This would provide an account of the ideas of Umemura and Malgrange, and their relationship with monodromy-preserving deformations & nonlinear pde.
Failing that, I'd settle for an updated edition of Pommaret's Differential Galois Theory.
A: 
    Mathematical theories deconstructed
  A compendium of dependencies by (and for) generalizers and formalizers

We know these theorems are proved. Now we want to know, which precisely pieces of foundations do they use. Does any of these inequalities … require ℝ, or an arbitrary ordered field is sufficient? For which an ordered commutative ring is enough? To which algebras/rings/manifolds can we generalize an analytic function … (although there are no power series)? Does the theory … effectively use the set theory, or it feels well with first-order logic? And with which namely? How different definitions of real numbers affect accessibility of theorems in analysis? What remains provable in topology without the law of the excluded middle? Which namely can be wrong in a theory of … for the statement … to become broken? On which exactly theories relies the best known proof of … theorem? And, generally, what is the mathematical truth?
Mathematics is a huge network of interdependencies but (in literary form) theories are ordered from postulates to conclusions, except for this book. Several nuances in definitions, not expressed before, are explored. And many unsolved problems “can we prove a well-know theorem … without assuming (allegedly true) statement …” are also listed.
A: Counterexamples in scheme theory
A: An English translation of Curtis and Reiner, Methods of representation theory with applications to finite groups and orders would be nice.
A: *

*"(Counter)examples in Algebraic Topology"

There are many good textbooks in homology and elementary homotopy theory, but the supply of instructive examples they offer is usually appallingly small (spheres and projective spaces are the standard examples, but often there is little beyond). One reason is that to discuss interesting examples, one needs a lot of machinery, whose development consumes time and space.
The books by Hatcher or Bredon offer a lot of examples; and I also like Neil Stricklands bestiary:
https://strickland1.org/courses/bestiary/bestiary.pdf,
and together with the unwritten chapter "things left to do", it is pretty close to what I would love to see as a book.
A: 
The Springer Correspondence

Tonny Springer developed a subtle correspondence between Weyl group representations (say over $\mathbb{C}$) and nilpotent orbits of the related semisimple Lie algebra,
showing in particular how to realize the finite group representations in the top cohomology of fibers in his special desingularization of the nilpotent variety.
By now the ideas involved have permeated much of the work in Lie theory due to Lusztig and many other people.    But there is no systematic treatise on the subject and its connections with other areas of Lie theory, algebraic geometry, combinatorics.   In my 1995 book Conjugacy Classes in Semisimple Algebraic Groups I included toward the end a very short survey of Springer theory, following a treatment of the unipotent and nilpotent varieties.   But I realized at the time that I didn't understand the subject deeply enough to write a comprehensive account.    (I still don't.)    
My first exposure to Springer's ideas unfortunately didn't take hold right away.  I recall making a short visit to Utrecht around 1975, where I had lunch with Springer at an Indonesian restaurant and he jotted down the new ideas he was excited about.  No napkin or other scrap of paper survives, but anyway I understood only later how amazing his insights were.   They deserve a thorough treatment in book form.
A: For a popular account an autobiographical Six Million Dollar Man: How I solved all six of the millennium problems in 1 year by anonymous author would definitely top my shelf.
On a bit more serious note, I am looking forward to...


*

*Continuum Hypothesis Part I and II with a chapter headed The Art of Forcing

*Five Pillars of Mathematical Logic (an encyclopedia in the same vein as the Russian EOM with 8000 entries from Logic only)

*On formalizing predicative notion: From zero to Γ0 in 2 seconds...

*Alan Turing's unpublished papers

*Ω: Absolute Infinity (perhaps this being sequel to Heller and Woodin edited Infinity)

A: Three views of differential geometry
I have in mind the most rigorous modern view, the most intuitive undergraduate calculus view, and the physicist's tensor calculus view. These perspectives can be so different that it's hard to keep in mind that they're all ultimately concerned with the same thing.  
Take one concept at a time examine it from a rigorous, intuitive, and computational viewpoint. For example, take a gradient and define it as a differential form, as a vector perpendicular to a surface, and as a tensor. Or here's how a differential geometer, a calculus student, and a physicist all view integrating over a surface. Here's how they each view Stokes' theorem etc. 
A: Spaces of Diffeomorphisms
For 60+ years this has been a foundation of differential topology, featuring prominently in work of Smale, Cerf, Hatcher, Thurston, and many others; but I don't know any adequate reference. Indeed, it seems only a handful of brilliant people know this stuff, and everyone else uses their work as if it were a collection of black boxes.
My dream book would include, among other things, a modern introduction to Cerf theory from the perspective of Igusa's theory of framed functions, leading up to a readable and self-contained proof of Kirby's Theorem. It would also contain exposition and simplification of theorems of Hatcher, Cerf, Kirby, and Seibenmann.
This is a cheerful prod to a certain prospective author of such a book, that when it is written it will surely become an instant classic; I, for one, will pre-order. 
A: As a counterpoint to gowers' answer, here's a different book -- or rather book series -- on physics which I'd like to see.
For the series, one would select a series of widely-used physics textbooks, and for each one, would write comprehensive mathematical footnotes, appendices, and references to more complete mathematical texts.
This way, one could skim the physics textbook at a first pass to get an idea of how physicists think about the subject matter, and then come back to the footnotes and appendices to get a sense for

*

*at which points there are mathematical details / inconsistencies being glossed over (e.g. boundary conditions, path space measures, existence & uniqueness of various representations, ...)


*what the larger context is for some of the methods being used (e.g. Hamlitonion / Lagrangian / Legendre transform, ladder operators and theory of weights, spectral theory, differential geometry, ...)


*where conventions differ between physicists and mathematicians (terminology and bases for Lie algebras, differential geometry, etc.)


*explanations and justifications of what is meant when a physicist says things like "the most general expression you can write down satisfying certain constraints (some of them implicit) is ____"


*etc.
For these purposes, I think it would be best to select physics textbooks which are minimally rigorous / truly introductory: in my experience with physics textbooks, often when they make some attempts at mathematical rigor, they just bog down the storyline and end up doing the mathematics badly anyway. I think it would be preferable to just see the physicist do things their own way, and when more mathematical aspects come up, to just punt them to an actual mathematician for further explanation.
A: The algebra, geometry, and combinatorics of Total Positivity
This is an area which has seen a ton of interest in the last 30 or so years, but there is no canonical textbook.
Possible topics include:

*

*the remarkable combinatorial/topological structure of totally positive spaces (e.g., https://arxiv.org/abs/math/0609764 and https://arxiv.org/abs/1904.00527);

*connections with canonical bases and cluster algebras (e.g., https://math.mit.edu/~gyuri/papers/pod.ps and https://arxiv.org/abs/1005.1086);

*applications in physics (e.g., https://arxiv.org/abs/1212.5605).

A: Galois representations. 
I know about Serre's Abelian $\ell$-adic representations and elliptic curves, but I am sure that a more general theory has been established since then. There are a few people who have notes on Galois representations on their web pages, but no book that I know of.
A: "Examples in complex geometry."
The algebraic and differential geometry and Hodge theory side of complex geometry is well established in many books, but I've had real trouble finding examples that are worked out in detail (which would be perfect as exercises, perhaps if given with hints) that show how the theory works in practise and provide counterexamples to some implications. For example, an ample line bundle does not have to admit any global sections, but I've never seen an example of such a bundle given in a textbook.
A: I would have killed for this a couple of years ago: a big book on Floer homology, written to be understandable for graduate students. Includes all the analytical details. 
A: *

*"Faltings explained" : Several of his articles are very hard to read and existing surveys on his concepts don't really fill the gap. I would like to read a book about his work, his themes, background ideas and techniques which is a readable walk through all that, something like Connes' "NCG"-book + Connes/Marcolli's "noncommutative garden".  

*"Morava explained" : The same as above on Morava's work, containing a (for the arithmetic geometry inclined reader) readable description of the homotopy theory background. With comments from Manin, Kontsevich and Connes, and a (sci-fi ?) chapter on how homotopy theory and number theory may mutually interfuse (e.g. through "brave new rings"). 

*Mumford suggested in a letter to Grothendieck to publish a suitable edited selection of letters by Grothendieck to his friends, because the letters he received from him were "by far the most important things which explained your ideas and insights ... vivid and unencumbered by the customary style of formal french publications ... express(ing) succintly the essential ideas and motivations and often giv(ing) quite complete ideas about how to overcome the main technical problems ... a clear alternative (to the existing texts) for students who wish to gain access rapidly to the core of your ideas". (Found in the very beautifull 2nd collection)  
A: You forestalled some of what I would have posted...


*

*Quillen's K-theory without topology

*Steenrod algebras through combinatorics and representation theory (as opposed to, through topology)

*Ext and Tor defined constructively, with Haskell code

*Weyl's "Classical Groups" with the proofs of 1938 but the notations of 2010

*The definitive guide to Hochschild homology

*Henri Lombardi's "Algèbre Commutative" in English

*A documentation to Agda
A: The construction of galoisian representations associated to primitive cuspidal eigenforms.  I hope the user BCnrd gets the hint.
A: A Bourbaki book on Category Theory.
A: Algebraic Geometry from a Homotopical Viewpoint: For the topologist who really wants to like geometry but doesn't know where to start.
A: As I have been telling many people involved in mathematical publishing, the one book I would like to read is The Serre-Tate correspondence.
A: Categories for the Working Mathematician
I know Saunders Mac Lane already wrote a book by that name, but in my opinion his book doesn't live up to its title. His book would perhaps be better named "Category theory for the working algebraist."  I'd like to see a book with more examples, especially examples outside of algebra and algebraic topology.
A: My answer is quite simple and stupid. I don't know French; so I would like to read EGA, SGA, and BBD in English (or in Russian:)). I also suspect that these books could be updated in the process of translation.:)
A: Algebraic groups by example
There are currently several books on Lie theory which take a very concrete approach, containing many examples (e.g. Rossmann, Hall, Stillwell). Basically they can be read by a student with some knowledge in calculus, linear algebra and perhaps some mathematical maturity. However, I have yet to find a book on the theory of (linear) algebraic groups which doesn't delve into topics from commutative algebra and algebraic geometry before even defining what an algebraic group is, and even then, most texts take a very abstract approach - most proofs seem like general nonsense to me, but maybe that's just because I'm not an algebraist in heart. In any case, I would very like to see a book on the subject which takes a very concrete approach through examples and constructive proofs.
A: Remark: Several items below refer to the formalism of locales.  Although consistent usage of the language of locales allows one to get rid of the axiom of choice in almost all cases, my main reasons for it are purely pragmatic: The formalism of locales allows one to obtain equivariant and family versions of many theorems without any additional effort, as opposed to the formalism of topological spaces (think of Hahn-Banach theorem, for example).

*

*A general topology textbook written in the language of locales, with no mention of topological spaces.


*Textbooks on commutative algebra and algebraic topology written in the language of locales.
In particular, such textbooks can usually avoid mentioning maximal ideals, the axiom of choice, or Zorn's lemma.


*A measure theory textbook written in the language of locales and commutative von Neumann algebras, with no mention of the set-theoretical approach.
The textbook should also have a conceptual exposition of Lp-spaces.


*A linear algebra textbook that does not mention coordinates, bases, or matrices.


*A textbook on smooth manifolds that never mentions coordinates, charts, or atlases.
Such a textbook should have a conceptual exposition of integration and use supermanifolds consistently whenever it makes sense, e.g., for differential forms.


*Textbooks on algebraic topology and homological algebra written in the language of (∞,1)-categories.


*Higher categories for the working mathematician.  This book should contain a lot of examples
of higher categories that are actually used in mathematics outside of category theory.
(For example, the bicategory of algebras, bimodules, and intertwiners,
the tricategory of conformal nets, defects, sectors, and morphisms of sectors etc.)


*A textbook on topological vector spaces (in particular, on locally convex, Banach, and nuclear spaces)
written from the categorical viewpoint.
For example, such a textbook would define a nuclear morphism as a morphism
that can be factorized in a certain way (see a recent paper by Stephan Stolz and Peter Teichner).
The textbook should consistently use the language of locales.  For example, this allows one to prove Hahn-Banach, Gelfand-Neumark, or Banach-Alaoglu theorems without using the axiom of choice.
A: Whittaker and Watson with a Facelift
There are a number of classic books, such as Whittaker and Watson's Modern Analysis, that I'd like to see typeset in TeX and updated slightly.  Sometimes notation or terminology have changed and a little footnote would help greatly.
Also by Watson, I'd like to see his 1922 book "A Treatise on the Theory of Bessel Functions" with updated typography and notation. A scan of the book is available here.  Apparently the book has entered the public domain and so there would be no legal barrier to producing an updated version.  
A: I would like to read a comprehensive, step-by-step introduction to the Langlands Programme written for non-experts. An Introduction to the Langlands Program (edited by Joseph Bernstein and Stephen Gelbart) is good, but it is a collection of articles, not a textbook or monograph. Stephen Gelbart's "An Elementary Introduction to the Langlands Program" (Bulletin of the AMS, Vol. 10, No. 2, 1984, pp. 177-219) has the right approach, but while quite long, is not a book-length treatment. David Nadler's excellent new article "The Geometric Nature of the Fundamental Lemma" is another example of the sort of expository approach I would like to see in a full-length book about the Langlands Programme.
A: 
(Counter)examples in scattering theory

A: Perhaps books along the lines of the Oxford series
of "very short introductions" to various topics.
This includes the masterful
Very Short Introduction to Mathematics by Timothy Gowers.
But one could imagine narrowly focussed, specific topics in mathematics,
perhaps no more than 150 pages, on many of the topics that have been suggested in
response to this post.
The closest similar suggestion is Mirco Mannucci's
"higher mathematics for the impatient."
Alas, unlikely to be commercially viable.
A: Pitfalls in infinite dimensional (differential) geometry
Nowadays there exists a plethora of results on Riemannian, symplectic and Poisson geometries on infinite dimensional manifolds. While classical texts cover much of the theory up to Banach spaces (e.g Langs Fundamentals of diff geom.) and some nice survey articles exist (the articles by Bruveris on Riem. Geom. for example) a comprehensive guidebook would be nice. Some of the material is of course covered in Kriegl, Michors big book on convenient calculus, but the frequent questions on MO on the topics (let alone the discussions I had with graduate students over the years) indicate that this topic deserves a book.
A: Somewhat frivolous/exasperated suggestion:
The Homology of Banach and Topological Algebras, Vol. II: Collected folklore and missing bookwork.
I only suggest this because I have been needing to cite this book, on and off, for much of the last five years, and the fact it's not been written hasn't really helped.
A: There are precisely two books on Arakelov geometry.  One by Lang and one by Soule. I would love to see a book written on the subject which focuses mainly on the two dimensional (and one-dimensional) case. Sections 8.3 and 9.1 of Liu's book do this greatly for example (but considers only intersection multiplicities at the finite points). It should include all the theorems done so far. Something like
Chapter 0. Prerequisites
Chapter 1. 
Arithmetic curves (Riemann-Roch, slopes method, etc. One should include a paragraph or appendix on algebraic curves stating all the theorems that can and have been generalized.)
(N.B. An arithmetic curve is the spec of a ring of integers.)
Chapter 2.
Arithmetic surfaces (This would contain all the "arithmetic" analogues of the theorems mentioned in the Appendix. For example, there has been a lot of work on Riemann-Roch theorems, trace formulas, Dirichlet's higher-dimensional unit theorem, Bogomolov inequalities, etc. Also, there are four intersection theories (which are compatible) I know of at the moment. The one developed by Arakelov-Faltings, then Gillet-Soule, then Bost and then Kuhn. The book should include a detailed description of them.
Appendix A.
Algebraic surfaces. (A survey of all the classical theorems for algebraic surfaces that have an analogue in Arakelov geometry. This includes Faltings' generalizations of the Riemann-Roch theorem, Noether theorem, etc. but also the theorems generalized to Arakelov theory by Gasbarri, Tang, Rossler, Kuhn, Moriwaki, Bost, etc.)
Appendix B.
Riemann surfaces (Just the necessary. Differential forms and Green functions basically.)
A: I'm surprised that nobody has expressed the desire to read Bourbaki's Théorie des nombres.
A: Introduction to algebraic cycles.
With lots of examples...
A: "Cardinal Arithmetic: The New Corrected Edition (including index)" by Saharon Shelah...
A: I would like to read an SGA-like book on Étale cohomology to replace as a reference SGA 4½. 
I also have an idea about who could write such a text: Luc Illusie. I'd really love that.
A: "The proof of the Shimura-Taniyama conjecture, for people who aren't professional algebraists but are willing to try pretty hard."
A: In pursuit of Hilbert's Problems
I think Hilbert's 23 problems form an organizatory framework for mathematics, that is much more organic than say the AMS classification. I believe that a book that traces the mathematics that grew from these problems can help to organize the burgeoning state mathematics is currently in.
I'm aware there is a book called "The Honours Class" that gives a history of Hilbert's problems up to their solution. However, this book is more biography than mathematics. Also, I'm interested in what happens after the problem is solved. A case study is the 17th problem, which lead to much of real algebra and real algebraic geometry today.
A: Algebraic Number Theory for Algebraic Geometers.
I want a book which explains algebraic number theory to somebody who is fluent in algebraic geometry (or maybe at least has read Hartshorne) but knows very little number theory to start with. In particular, such a book would not be seeking to minimize the prerequisites required of the reader, but rather adapt itself to a very particular set of prerequisites. At a bare minimum, some of the peculiar terminology of number theory -- "conductor", "discriminant", ... -- would be defined in terms of algebro-geometric concepts.
I asked a question about this here. Somebody suggested Neukirch's book, which is better than most in this regard, but even Neukirch has a different audience in mind, and confines his algebro-geometric discussions to impressionistic side-chapters rather than integrating them fully into the text.
A: The book I would love to read (and own) is called:
HIGHER MATHEMATICS FOR THE IMPATIENT. (HMFTI)
Description: the book appeals to all math aficionados who have at least a solid ground in basic advanced math (say at the level of standard qualifying math department examinations in the USA).
Now, it is sort of similar to the Princeton Companion of Mathematics, but without the biographies, and in which the articles are full-fledged chapters, with a core introduction (main idea, main results) and a LOT of  worked out key examples (take note of the word key, I would like to have only examples which help me grab the essence of the field, and nothing else).
So, just to give you a flavor, take Algebraic K-theory. At some point I kinda knew what it is, but I would love to grab HMFTI, get myself a glass of brandy, a churchwarden pipe, a notebook and a pen, and read the chapter, doing the exercises just enough to have a full sense of the field. Same for the other chapters, say Simplectic Geometry, Orbifolds, Finite Geometry,  Higher Categories, etc.
PS This book would be -I think- REALLY nice, but ain't easy to write, in fact extremely hard. You know why? Because it is much much easier writing a text in a field stuffed with all the latest results than a brief introduction which conveys the essentials and nothing but the essentials, all the while being no popularization
A: I don't know for certain that this doesn't exist, so I'm in a no-lose situation: if this is a rubbish answer then it means that a book that I want to exist does exist. Many mathematicians of a pure bent have taken it upon themselves to get a good understanding of theoretical physics. And many have actually managed this. But it seems to me that they usually go native in the process, with the result that I cease to be able to understand what they are saying. It could be that this is just an irreducibly necessary feature of physics, but I doubt it. Out there in book space I believe there exists a book that explains theoretical physics in a way that physicists would dislike intensely but mathematicians would find much easier to read. It may well be that if you want to do serious work in mathematical physics then you have to understand the subject as physicists do. However, this book would be aimed at pure mathematicians who were not necessarily intending to do serious work in mathematical physics but just wanted to understand what was going on from a distance.
I used to have a similar view about explanations of forcing, but I think Timothy Chow's wonderful Forcing for Dummies has filled that gap now.
A: "Quiver varieties with a wealth of examples" ?
A: C.P. Snow once used such persuasion as he had to get G.H.Hardy to write another book, which Hardy promised him to do. It was to be called 'A Day at the Oval' and was to consist of himself watching cricket for a whole day, spreading himself in disquisitions on the game, 
human nature, his reminiscences, life in general. Unfortunately Hardy's final years of his life were not of delight and the book, though destined to be an eccentric minor classic was never written.
I would love to see such a book, written with incomparable style and mathematical touch.
A: Variational PDE's and Lagrangian field theory for mathemticians
Also
Hodge theory of asymptotic expansions of oscillatory integrals and resurgence
A: I, as an undergraduate student in physics, would really like a comprehensive solutions book for Roger Penrose's The Road to Reality: a complete guide to the laws of the universe (Vintage, 2004)
A: I think for graduate courses, it would be nice to have a graduate textbook on "stochastic partial differential equations" at the same exposition level as the usual graduate pde's textbook by Evans. So many examples and lots of instructive exercises.
For example, as mentioned here: Good books on stochastic partial differential equations? the main one is the Stochastic Equations in Infinite Dimensions by da Prato and Zabczyk. But it doesn't have any lists of exercises.
There are many good monographs eg. the one by Hairer An Introduction to Stochastic PDEs.
I also like the book "a course on rough paths" which goes into SPDEs too.
