In my experience it can take as little as a single bone, what's the maximum it could possibly be?
There is a 1 in 3 chance any particular bone will tame a wolf.
If you have a stack of
x bones, the chance you will be able to tame a wolf before running out of bones is
1 - ((chance of not taming with bone) ^ x) 1 - ((2/3)^x)
Fill in for
x to get your chances.
For a 99% chance to get a wolf, you will need 12 bones. 17 bones will get you 99.9% chance, and 22 bones will get you up to 99.99% chance.
NOTE: I got the 1/3 figure from http://minecraft.gamepedia.com/Wolf -- in the same sentence, it also says "up to 12 bones need to be used to ensure that the wolf is successfully tamed." This sentence cites a tweet from jeb which makes neither of these assertions, although the tweet does confirm that it is a random chance. I'm running with the 1/3 figure and assuming they are just assuming nobody will hit that 1% chance, although it's a virtual certainty that some players have gone into the 20 bone range.
The 1/3 figure matches my anecdotal experience, and it's the only place I've found that attempts to give a solid number on it. Probability is a subject that trips up a lot of people, so a lot of results come back with people giving answers with an authoritative tone that seem unsupported (e.g. saying it's random and then guaranteeing an upper limit to the number of bones needed).