Using a randomized algorithm to estimate it should be reasonably cheep. Also compute per is very cheep these days. The benefit would be twofold:
1) Curiosity - In the same way that people want to know how closely they are related to the Queen of England, I think people would be curious how well they were connected to the average person.
2) To give some measure of progress a person has made in connecting themselves (or others) more fully into the global tree. I'm new here, but I feel the community would like the global tree as fully connected as possible. It would be nice to have some way to measure that.
Finally, I just personally love data and math and so would be interested in the mathematical structure here. I was curious if anyone else shared my interest.