Keep rolling while you repeat the first die value

Roll a fair die once. Then keep rolling, stopping at the first roll whose value is different from the first roll's value. You are paid the total of every roll made, the first and last included. What is the fair price of this game?

$7.7$. Given a first roll of $i$, the number of repeats of $i$ before a new value is geometric with mean $\tfrac15$, and the final roll is uniform on the other faces with mean $\tfrac{21-i}{5}$. So the conditional total is $i+\tfrac i5+\tfrac{21-i}{5}=i+\tfrac{21}{5}$, and averaging over $i$ gives $3.5+4.2=7.7$.