The fact that Hochschild cochain complexes are not commutative, but only commutative up to a controlled homotopy was a fundamental insight of Gerstenhaber.There is moreover a series of higher operations that can be neatly organized into an operadic structure, which is the content of Deligne’s conjecture.There is a whole package of operations on the chain level based on surfaces with extra structure, which we gave in 2006.In the case of a Frobenius algebra this yields a homotoy BV structure that essentially captures a circle action. In geometric situations this captures algebraic string topology operations, for instance  an operation corresponding to the Goresky-Hingston coproduct.
In newer work with M. Rivera and Zhengfang Wang, we extend these operations to Hochschild chain complexes and the Tate Hochschild complex,which connects the chain and cochain complexes and plays a role in singularity theory.
There is a particular dualization which dualizes a higher multiplication found in this complex toa double Poisson bracket. These operations are natural from the point of view of surfaces and also allow for animation in the sense of Nest and Tsygan.