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?
Hochschild (co)homology of (curved) A-infinity algebras/categories
Relation to deformation theory
Hochschild-cyclic spectral sequence and relation to Hodge-de Rham spectral sequence
Relation to Drinfeld center
2D TQFTs (Costello/Kontsevich/Hopkins-Lurie)
In addition to Kevin's excellent list:
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 personal list.
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.
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).