Answer

Fair value of the game is the expected value we'd receive from playing it. We calculate this as the sum of possible maximum values (1-6) times their probabilites. 6 is the max if first die is 6 and second die is anything 1-5 (5 outcomes); if first die is anything 1-5 and second is 6 (5 outcomes); and if both are 6 (1 outcome). 11 in total: two symmetric cases where one die is the maximum value and other is strictly smaller, and one case where they both have the max value. Similarly, 5 is max in 9 cases; 4 in 7 cases and so on. The pattern is: n is the maximum of the two in 2n-1 ways. Each combination is equally likely, and there are 36 of them in total (6 possibilities for first die $\cdot$ 6 possibilities for the second).

Summing up all the outcomes multiplied with their probabilities, we get $EV = \sum_{n=1}^6 \frac{1}{36} \cdot (2n-1) = \frac{161}{36}$