Maximum Sum SubArray - Naive Implementation
An implementation that I wouldn't call naïve but criminally stupid would calculate the sum of elements in each subarray by adding up the elements. You are summing from 1 to n elements for each of n^2/2 subarrays, which makes it $O (n^3)$ (proving that this is a lower bound needs a bit more care). 

Of course a naïve but not stupid implementation would calculate the sums for all subarrays starting with the first element using only one extra addition for each sum.