Another valid answer:
Fun fact, coming back to this. This sequence will hit a point where it will output the same number forever.
A Fringe Word™
A Not Fringe Word™
A Leaping Set™
A Not Leaping Set™
A Persistent Phrase™
A Not Persistent Phrase™
A Square Number™
A Not Square Number™
A Double Jump Word™
A Not Double Jump Word™ has
So what makes an Interesting Puzzle™?
If we look at the answer words...
And what makes a not-Interesting Puzzle™?
OEIS doesn't list this sequence. After analyzing the pattern, I come to the conclusion that (one) answer is
EDIT (05/06/18): I submitted this sequence to OEIS and it has (finally) been approved now.