For consecutive numbers the average equals the middle value, and the sum equals the average times the count. Here the average is $108/8 = 13.5$. With 8 numbers, this average falls exactly between the 4th and 5th numbers in the sequence, so the 4th number is 13 and the 5th is 14. Building outward gives the full list 10, 11, 12, 13, 14, 15, 16, 17. The largest value is the last one. $\boxed{17}$