The average (arithmetic mean) of $10$ different positive whole numbers is $10$.  The largest possible value of any of these numbers is
$\text{(A)}\ 10 \qquad \text{(B)}\ 50 \qquad \text{(C)}\ 55 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 91$
If the average of the numbers is $10$, then their sum is $10\times 10=100$.  
To maximize the largest number of the ten, we minimize the other nine.  Since they must be distinct, positive whole numbers, we let them be $1,2,3,4,5,6,7,8,9$.  Their sum is $45$.
The sum of nine of the numbers is $45$, and the sum of all ten is $100$ so the last number must be $100-45=55\rightarrow \boxed{\text{C}}$.