What is a noncommutative hypercommutative algebra?

Hypercommutative algebras are algebras over the homology of the operad of Deligne-Mumford moduli spaces of genus zero curves with marked points. I shall report on joint work with Sergey Shadrin and Bruno Vallette, explaining a parallel story where underlying associative algebras become noncommutative. The role of Deligne-Mumford spaces in this story is played by toric varieties of (certain lattice realisations of) Stasheff associahedra.