How quickly does sugar cane grow?
Solution 1:
I 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.
Solution 2:
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/Tick