Source: Jane Street Capital Interview (Quantnet)
You are given a 100 sided die. After you roll once, you can choose to either get paid the dollar amount of that roll OR pay one dollar for one more roll. What is the expected value of the game?
(There is no limit on the number of rolls.)
Update (22 June 2014):
Solution posted by me, Felix, JDGM, Sebastian, Fu Shi and Dumb Phoenix in comments!