Sure -- try the set of homomorphisms from pi_1^{et}(E - O) to a fixed finite group G. This is a "non-abelian level structure" of the sort considered by de Jong and Pikaart
http://arxiv.org/abs/alg-geom/9501003
Rachel Davis, a 2013 Wisconsin Ph.D. working with Nigel Boston, wrote her thesis about this kind of stuff in the case of elliptic curves.