We'll never settle what's the busiest route in the world, but let's try to nail down which airport-pairs have the most flights.
These totals from http://www.oag.com
might be off by one or two due to undetected codeshare duplication, but hopefully these are close to the correct numbers of nonstop jet schedules (on 12 Aug) from
SDU to CGH: 88. Surely no other airport-pair can beat that? Or ever could?
TSA to KHH: 61. Current second place?
SYD to MEL: 55. Current third place? (Getting shaky now.)
HND to CTS: 49. And almost all widebodies, unlike the others.
HNL to OGG: 46, or 47 if you include the nonstop Dash 8. Dunno if this is fifth in the world, but it must be first in the US.
TPE to HKG: 37. Another all-widebody passenger/seat-count contender.
MAD to BCN: 40. Is that the top airport-pair in Europe? (LHR-DUB: 23)
Note that all the above are one-way totals. Seems to me that around 1990, before Southwest glommed onto Oakland, SFO-LAX had 75 flights, but now they're down to 30. Other US airport-pairs like LGA-ORD (32) have never had particularly impressive counts.