Book on Hochschild (co)homology There is currently no treatise treating Hochschild (co)homology systematically. There is a chapter in Weibel's book, there's parts of Loday's and a few others...

What should be covered by such a mythical treatise?

 A: I am working on producing an account from a modern perspective at Hochschild cohomology on the $n$Lab.
Some of the wishlist items expressed here are already being covered to some extent. But clearly more needs to be done.
A: I really want to know more about Hochschild cohomology of commutative algebras, and its relation to the representations of $S_n$ and to free Lie algebras (beyond "there are these strange idempotents in $\mathbb Q\left[S_n\right]$ which happen to occur in both fields).
A: Sarah Witherspoon's book on Hochschild cohomology for Algebras is now published. See https://www.ams.org/publications/authors/books/postpub/gsm-204
A: There is so much on this subject coming from different perspectives. 
 I would add to all of the above  a complete, up to date, exposition regarding the relationship of Hochschild homology with loop spaces (for example including the non-simply connected case) and string topology (the structure of the hochschild chain complex of an algebra with a homotopy version of Poincaré duality, etc...) 
A: In addition to Kevin's excellent list:
Formality
The relation of Hochschild and cyclic homology with loop spaces (eg Jones' theorem)
and the circle action on Hochschild homology
operadic structure of $(HH^\ast,HH_*)$ (ie "calculus" a la Tsygan-Tamarkin), in particular 
the BV structure in the Calabi-Yau case
Relation to the cotangent complex/ Andre-Quillen homology in the commutative case
The role of Hochschild homology as recipient of characters (eg Chern characters and characters of representations) -- more generally the relation with algebraic K-theory
topological Hochschild and cyclic homology, the cyclotomic trace, $K^S=THH$
HH for E_n algebras and the Deligne-Kontsevich conjecture
Lie theoretic perspective ($HH^\ast$ as universal enveloping algebra of the Atiyah bracket on the shifted tangent complex, HKR theorem as PBW, $HH^*$ as the Lie algebra of autoequivalences of the derived category...)
A: My wishlist:
Hochschild (co)homology of (curved) A-infinity algebras/categories
Relation to deformation theory
Hochschild-Kostant-Rosenberg
Hochschild-cyclic spectral sequence and relation to Hodge-de Rham spectral sequence
Deligne conjecture
Relation to Drinfeld center
2D TQFTs (Costello/Kontsevich/Hopkins-Lurie)
A: A personal list.


*

*Hope the hypothetical author would not restrict the base to be a field, even perhaps allowing another DGA as base.  (Most of the following comments probably assume this.)

*The flatness hypotheses over the base and Shukla homology as a repair for this.  Invariance of Shukla homology under weak equivalences of DGAs.

*Relationship to TorR⊗Rop(R,M).

*For R→S a map of commutative DGAs with S flat over R, the Hochschild homology spectral sequence starting with HHR(S, TorR(M,N)) and converging to TorS(M,N).

*Base-change formulas, ranging from simple ones to the following:
$$
S \otimes_{HH^R(S)}^{\mathbb L} HH^R(A,M) \simeq HH^S(A,M)
$$

