Scott, I have often wondered that myself. Well I found a good answer. The answer depends on two things, first, who is licking the tootsie pop. If you have acidic saliva it would take fewer licks to get to the center. The second thing is, it depends on what type of sugar they use in the tootsie pop.
Some sugar disolves faster than others.
The best way for you to find you answer is to try licking several tootsie pops and see what the average is. It probably wont be the same each time so you may have to guess somewhere in the middle. Tell me what you find out when you do your experiment.