Largest possible sample value

Make the other numbers as small as the constraints allow.

1. The total is $5\times 6=30$.
2. Sorted, the numbers are $a<b<6<d<e$, since the median is $6$.
3. To maximise $e$, take the smallest options $a=1$, $b=2$ and $d=7$.
4. Then $e=30-1-2-6-7=14$, so the answer is $14$.

Answer: $14$