As usual, you have to decide what "busiest route" means. Does "route" mean one airport-pair? Or if we group airports, we have to decide how large the groups can be.
Then you have to decide whether you're counting flights or passengers, and, if the latter, maybe you have to decide whether you're counting O&D passengers or all passengers.
Busiest airport-pair, by passenger count: I'm guessing Tokyo Haneda to Sapporo; the seat count is probably highest there?
Busiest airport-pair, by flight count: Santos Dumont to Congonhas used to be tops, and maybe still is; Madrid-Barcelona is another possibility. Taipei-Kaohsiung (?) wouldn't be too far behind, along with other Asian routes. No US route can compare any more, though circa 1990 LAX
had about 90 each way a day, which may be the all-time record. Sydney-Melbourne was heavy for a time but I guess they're down now?
No London-Dublin airport-pair is a contender, and when you group their airports I suspect they have a hard time beating Los Angeles area to San Francisco area-- by flight count anyway.
As to the original question-- if by "Los Angeles to San Francisco" you meant LAX-SFO, that airport-pair hasn't been the busiest airport-pair for years, if it ever was.
[Edited 2004-06-03 23:57:30]