Expected maximum roll before stopping on a 4

Jim keeps rolling a fair six-sided die and stops the first time a 4 comes up. What is the expected value of the largest number shown over all of his rolls, including the final 4?

$31/6$. The maximum is 4, 5 or 6, and only the relative order of first appearances of 4, 5, 6 matters. A 6 precedes the 4 with probability $\tfrac12$; the max is 5 only for the order 5, 4 (with 6 after), probability $\tfrac16$; otherwise it is 4, probability $\tfrac13$. So $E=4\cdot\tfrac13+5\cdot\tfrac16+6\cdot\tfrac12=\tfrac{31}{6}$.