As can be proven, Maxwell's equations don't work for spherical or cylindrical waves, but these waves satisfy the wave equation. Are there any other examples of this type that satisfy the wave equation but not Maxwell's equations?
Mmm, the wave equation is a consequence of Maxwell's equations plus a gauge condition, and once you cast the wave equation into spherical, or cylindrical coordinates, you can easily find solutions that satisfy it. Maybe you mean waves that are spherically symmetric? In that case, the problem is more about finding non-trivial spherically symmetric vector fields (you can't) and nothing to do with Maxwell's equations. However, the waves can have a spherical character to them, as is the case with dipole radiation.