Longest Increasing Substring
Pyth, score 2 (8 bytes)

lefSIT.:


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Code points [108, 101, 102, 83, 73, 84, 46, 58]. Another shorter solution, leSI#.: scores 3, but its code points are [108, 101, 83, 73, 35, 46, 58], which are very close to a score of 1, actually. Rearranging a bit may help Nevermind, the substrings built-in is .: which cannot be rearranged, so the lowest score must be 2 if the program makes use of it.

How?

lefSIT.:     Full program. Accepts either a list or a string from STDIN.
      .:     Substrings.
  f  T       Only keep those that are...
   SI        Sorting-Invariant.
le           Length of the last item.