Die rolls that continue on even faces and heads

A game uses a fair six-sided die and a fair coin. Each time you roll the die you are paid the number shown. An odd roll ends the game immediately. After an even roll you toss the coin: tails ends the game, while heads sends you back to roll the die again, with the same rules applying to every later roll.

What are your expected total winnings?

$\tfrac{14}{3}$. Condition on the first roll. An odd roll pays $3$ on average and stops; an even roll pays $4$ on average and then, with probability $\tfrac12$, the whole game restarts. So $E=\tfrac12\cdot 3+\tfrac12\left(4+\tfrac12 E\right)$, giving $E=\tfrac{14}{3}$.