As it took me about 2 years, off and on, to complete reading this tour de force, I would appear to have demonstrated Hofstadter's Law: "It always takes longer than you expect, even when you take into account Hofstadter's Law."
It is difficult to provide a summary of the subject of the book: it's about mathematics, but not the "everyday" arithmetic that most people would think of when you said "mathematics", and it's also about philosophy. The first few chapters introduce the notion of Formal Systems, and the remainder of the book explores the bizarre consequences of Gödel's Theorem regarding their incompleteness. Much of the material focuses on what happens when some set of rules (a system) is applied to itself: this perhaps offers an explanation of the presence of a number of M C Escher works in the book, which often present visual paradoxes and dilemmas, and the references to J S Bach, the undisputed master of self-referential fugues.
Between each chapter in the book there is a "dialogue". A number of unlikely characters (mainly Achilles, the Tortoise, and the Crab) discurse into various matters, sometimes closely related to the main themes of the book, sometimes more obtusely so. As Hofstadter himself points out, in many places the dialogues are so thick with meaning on so many levels it is impossible not to re-read certain passages several times. The writing of them must have been tortuous, but undoubtedly justified by the achievement of such spectaculars as Contracrostipunctus, the Crab Canon and the Prelude ... Ant Fugue. The subtitle "A metaphorical fugue on minds and machines in the spirit of Lewis Carroll" seems appropriate indeed.
As Hofstadter peruses formal systems, he encounters the principles of computer languages. This leads to considerations of self-reference, self-replication, DNA, and speculation on the origin of human intelligence.
It is not necessary to understand every concept presented in the book to enjoy it, though obviously the more that is understood the greater will be the enjoyment, just as is the case with the many meanings in the dialogues. Although the book is "mathematical", it's not the kind of maths that you learn at school, and an aptitude or otherwise for this should not preclude reading the book. The only parts of the book that have dated since its original publication in 1979 (my copy is the 20th anniversary edition) are some sections on artificial intelligence (specifically, the assertion that no computer program exists that can beat a chess grand master is obviously no longer the case), but this in no way detracts from the whole.
It's a book that I will have to re-read, ideally without so many diversions.