Atalanta, uninterested in marriage, agreed to marry only if her suitors could outrun her, though fully armed, in a footrace. King Schoineus agreed and many young men died in the attempt until Melanion (or Hippomenes) came along. Melanion asked the goddess Aphrodite for help and she gave him three golden apples to toss as Atalanta caught up, in order to slow her down.
If it can be called a "weakness"; then the golden apples of Aphrodite.
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Atalanta was a heroine in Greek mythology.
Penelope, the wife of Odysseus.
Nemesis is a goddess. No myth involving a particular weakness, either.
Not that is known in mythology.
He can be affected by his own arrows.