How quickly does sugar cane grow?
Sugar cane grows using block ticks. Every 16 block ticks it grows, and each block tick happens on average every 68 seconds, that means on average a sugar cane will grow every 1088 seconds, or 18 minutes 8 seconds. This is only with optimal conditions however. Watch all of Wubbi's video, it's really helpful and accurate.
Sources:
http://minecraft.gamepedia.com/TickI have an automatic piston-harvested farm with 16 cane plants harvested every ~10 seconds. Here are two different estimations of the rate. By "growth" I mean the event of one block being added to the top of a sugar cane plant.
When I watch the farm running, it seems to drop about 1 cane every 2 cycles (but this is a fuzzy memory); this would be a rate of (1 cane)/(20 seconds)/(16 plants) = 1/320 growths per plant per second.
When I come by to collect its drops, I usually get 6-8 canes (but this excludes the number which miss the water stream and get stuck inside the farm); if the farm has been active and the collection point untouched for a full 5 minutes = 300 seconds (item despawn time), then we get (8 canes)/(300 seconds)/(16 plants) = 1/600 growths per plant per second.
The latter figure is probably closer to correct as it does not involve an estimation of probability.