Statistically, far, far fewer heroes have archenemies than they ones who do. If you really do the calculations, probably less than 1% of DC’s superheroes have them.
Yeah Lady Shiva would be closest (other than that time she joined the League which imma pretend didnt happen). Her father, David Cain would also be another, but their relationship is more complicated. At least in NE.
Lady shiva is her problem but she is no batman or superman she is trained to guard ras al ghul so she may be greatest fighter than batman and ladushiva but she is no vengeance like batman hope like superman and she is not speed like falsh. But she has her role as a good and very skilled fighter of dc universe. May be she doesn't need a archenemy to decide her power and skill.