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)