My wishlist:
Hochschild (co)homology of (curved) A-infinity algebras/categories
Relation to deformation theory
Hochschild-Konstant-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)