Great rivalies can be measured in different ways. Longevity is probably one of the top barometers of greatness, but can be tainted by disminished competetiveness.

For Longevity:
Yankees - Boston (also mystique)
Army - Navy (national pride and tradition)
Harvard - Yale (probably the oldest).

For Intensity:
Ali-Frazier
Connors-Borg

Whoever exalts himself will be humbled, whoever humbles himself will be exalted - Matthew 23:12