  Quasi-characters are vector-valued modular functions having an integral, but
not necessarily positive, q-expansion. Using modular differential equations, a
complete classification has been provided in arXiv:1810.09472 for the case of
two characters. These in turn generate all possible admissible characters, of
arbitrary Wronskian index, in order two. Here we initiate a study of the
three-character case. We conjecture several infinite families of
quasi-characters and show in examples that their linear combinations an
generate admissible characters with arbitrarily large Wronskian index. The
structure is completely different from the order two case, and the novel coset
construction of arXiv:1602.01022 plays a key role in discovering the
appropriate families. Using even unimodular lattices, we construct some
explicit three-character CFT corresponding to the new admissible characters.
