An edge-to-edge geodetic set S in a connected graph G is called a minimal edge-to- edge geodetic set if no proper subset of S is an edge-to-edge geodetic set of G. The upper edge-toedge geodetic number gee +(G) of G is the maximum cardinality of a minimal edge-to- edge geodetic set of G. The upper edge-to-edge geodetic number gee +(G) of a graph is studied and is determined for certain classes of graphs. It is shown that, for every pair a, b of integers with 2≤a≤b, there exists a connected graph G such that gee(G)=a and gee +(G)=b.