If sorting is not desirable, you can use a combination of hash map and Disjoint-set data structure.
For each element in the list create a node and insert it into hash map with key = element's value. Then query the hash map for value+1 and value-1. If anything is found, combine current node with set(s) where adjacent nodes belong. When finished with the list, the largest set corresponds to the biggest interval.
Time complexity is O(N * α(N)) where α(N) is inverse Ackermann function.
Edit: Actually Disjoint-set is too powerful for this simple task. Solution by Grigor Gevorgyan does not use it. So it is simpler and more efficient.
