This paper surveys various kinds of ordered sets, with numerous citations to sequences in the On-Line Encyclopedia of Integer Sequences. These ordered sets include self-containing sequences, infinitive sequences, fractal sequences, and parasequences (which are introduced here as a certain type of doubly infinite sequence). Relationships among these are presented, and among more than thirty examples, the Cantor fractal sequence and the Farey fractal sequence are presented. There are several conjectures involving parasequences.