The maximum distance between two ellipses is not the same as the maximum distance between all pairs of points on them right? Like just imagine measuring the maximum distance from a circular orbit to itself. One measure would give zero (zero separation between the orbits) and the other would give twice the radius (individual points can be 2R away from each other).

If I'm doing this correctly I think the max distance between two points would come out to max_x max_y ||x - y|| whereas the max distance to the orbit would come out to max_x min_y ||x - y||.

There are orbital resonances, but you are right that none involve planets. https://en.wikipedia.org/wiki/Orbital_resonance

The closest is the 2:3 ratio of Pluto:Neptune. (It counts if you consider Pluto a planet. ;)

Otherwise, there's a nice table showing "near-integer-ratio relationships between the orbital frequencies". If I understand it correctly, each of the bigger-than-a-dwarf-planet planets will be at their furthest possible distances from each other in timescales no longer than about 50,000 years.

See here: https://news.ycombinator.com/item?id=16211745