  We introduce a class of iterated integrals, defined through a set of linearly
independent integration kernels on elliptic curves. As a direct generalisation
of multiple polylogarithms, we construct our set of integration kernels
ensuring that they have at most simple poles, implying that the iterated
integrals have at most logarithmic singularities. We study the properties of
our iterated integrals and their relationship to the multiple elliptic
polylogarithms from the mathematics literature. On the one hand, we find that
our iterated integrals span essentially the same space of functions as the
multiple elliptic polylogarithms. On the other, our formulation allows for a
more direct use to solve a large variety of problems in high-energy physics. We
demonstrate the use of our functions in the evaluation of the Laurent expansion
of some hypergeometric functions for values of the indices close to half
integers.
