  We study an elliptic analogue of multiple zeta values, the elliptic multiple
zeta values of Enriquez, which are the coefficients of the elliptic KZB
associator. Originally defined by iterated integrals on a once-punctured
complex elliptic curve, it turns out that they can also be expressed as certain
linear combinations of indefinite iterated integrals of Eisenstein series and
multiple zeta values. In this paper, we prove that the $\mathbb{Q}$-span of
these elliptic multiple zeta values forms a $\mathbb{Q}$-algebra, which is
naturally filtered by the length and is conjecturally graded by the weight. Our
main result is a proof of a formula for the number of $\mathbb{Q}$-linearly
independent elliptic multiple zeta values of lengths one and two for arbitrary
weight.
