Solution:

The sum is $8\cdot 8=64.$ The collection $6,6,6,8,8,8,8,14$ has mean, median, unique mode, and range all equal to $8,$ so $14$ is attainable.

If the largest were $15,$ the range $8$ forces the smallest to be $7,$ so all eight integers are at least $7.$ The other seven then sum to $64-15=49=7\cdot 7,$ forcing every one of them to equal $7.$ But then the median and mode would be $7,$ not $8,$ a contradiction.

Thus, the correct answer is D.