In a seminal paper "On the Deformation of Rings and Algebras", M. Gerstenhaber showed that the deformation complex of any associative algebra (known as the Hochschild complex) is naturally endowed with a structure of (later called) Gerstenhaber algebra (consisting of a degree 0 commutative associative product together with a compatible degree -1 Lie bracket).
The notion of Gerstenhaber algebra can be naturally extended to one of a $\hat p$-Gerstenhaber algebra where the Lie bracket is of degree $1-\hat p$.
$1$-Gerstenhaber algebras are associative algebras.
$2$-Gerstenhaber algebras are usual Gerstenhaber algebras.
In math/0010072, Tamarkin generalised the result of Gerstenhaber by showing that the cohomology of the deformation complex of a $\hat p$-Gerstenhaber algebra is naturally a $\hat p+1$-Gerstenhaber algebra (rather he showed that there is a homotopy $\hat p+1$-Gerstenhaber algebra structure on the deformation complex itself).
In particular, the cohomology of the deformation complex of a Gerstenhaber algebra is naturally a $3$-Gerstenhaber algebra.
Question: Are there other known structures whose deformation complex admits a $3$-Gerstenhaber algebra structure on cohomology?