Knapsack up to the heaviest item
The problem (are there items such that all except the largest fit, and the total value is ≥ V) is obviously in NP. It is also NP complete because we can use it to solve Knapsack: 

Given a knapsack problem, add an item with sufficiently large weight w, and sufficiently large value v, solve the problem with V + v, and the solution with the large item removed is the solution for the knapsack problem. 

We can also solve this problem using knapsack: For each item with weight w and value v, solve knapsack removing this item and all items with weight > w and a goal for the value of V - v.