(An operad that describes a multiplication that is associative but not necessarily commutative "up to homotopy" is called an A∞-operad.)
An operad A is said to be an E∞-operad if all of its spaces E(n) are contractible; some authors also require the action of the symmetric group Sn on E(n) to be free.
More generally, there is a weaker notion of En-operad (n ∈ N), parametrizing multiplications that are commutative only up to a certain level of homotopies.
The most obvious, if not particularly useful, example of an E∞-operad is the commutative operad c given by c(n) = *, a point, for all n. Note that according to some authors, this is not really an E∞-operad because the Sn-action is not free.
The operad of little n-cubes or little n-disks is an example of an En-operad that acts naturally on n-fold loop spaces.