[kadoban]

If Big O is always worst case, you're going to have a hard time saying anything interesting or useful about quicksort, for instance, or hash tables.

You're going to be limited to "quicksort is O(n^2)" and "hash table lookup is O(n)", both of which are quite misleading.

There are good reasons that we usually look at the worst case, but big-O does not fundamentally or necessarily mean that.