Are there examples of brace algebras that are not operads?

Question

The most typical example of a brace algebra is the brace algebra structure on the Hochschild complex of an associative algebra. This is a particular case of the following construction applied to the endomorphism operad of the algebra: for an operad $\mathcal{O}$ with composition map $\gamma:\mathcal{O}\circ\mathcal{O}\to\mathcal{O}$ one can define a brace algebra by taking $x\{x_1,\dots, x_n\}=\sum \pm \gamma(x;1,\dots, 1,x_1,1\dots,1,x_n,1\dots 1)$.

Are there any interesting examples of brace algebras that are not (in an obvious way at least) the result of the above general construction?

Answer 1

To record my answer in comments properly: brace algebras coming from operads satisfy one obvious constraint: for every $x$ and sufficiently large $n$ we have $x\{x_1,\ldots,x_n\}=0$, since we cannot plug more elements than the maximal arity of elements appearing in $x$. This fails in a lot of brace algebras, for example in free ones.