Yeah, this group isn't all that fun. But when King Pelias kicked Jason's dad off the throne, Jason's mom had to hide him for his own good. Jason had it a little easier than a lot of the other kids in this clique. Instead of being abandoned on mountain somewhere, he was raised and taught by Chiron, the gentle centaur.
These twins were tossed into the River Tiber along with their mom, Rhea Sylvia, by their evil uncle Amulius. Their uncle's dastardly scheme didn't work out, though, because the river god, Tiberius, spewed the boys on shore and married their mom. Later the twins were suckled by a she-wolf, fed by a woodpecker, and eventually taken in by a shepherd. Oh, and Romulus ended up founding Rome. (Not too shabby.)
Paris's parents assigned a shepherd the job of abandoning him on a mountain. They may sound kind of mean, but they only got rid of him because it was prophesied that he would eventually cause the destruction of his hometown of Troy (which he actually did end up doing). Paris survived his "exposure" on the mountain by being suckled by a friendly she-bear, who kept him alive until the guilty shepherd came back and raised him as his own son.
Just like Paris, Oedipus was abandoned by his parents because of a prophecy. In his case, the Oracle of Delphi said that little baby Oedipus would grow up to kill his father and sleep with his mother. So his parents did what any sensible parents would do; they pinned his ankles together and ordered a slave to abandon him on a mountain. The slave didn't have the heart to do it, though, and instead passed the little bundle of trouble off to a shepherd. Unfortunately for pretty much everybody involved, Oedipus grew up and unknowingly fulfilled the prophecy. (For the record, we're kind of glad there aren't too many prophecies any more.)
One of the toughest ladies in Greek mythology, Atalanta was abandoned by her father King Iasus because he wanted a son. Just like Oedipus and Paris, she was left on a mountain to die. Also like Paris, she survived because she was suckled by a she-bear. (Man, there were a lot of overly friendly she-bears back then.) Eventually, Atalanta was taken in by some hunters, and she became one of the best huntresses to ever string a bow.
This hero of heroes is the son of Zeus and the beautiful mortal woman, Alcmene. His mom was more than a little freaked out when he was born, because Zeus's wife, Hera, was pretty famous for being the jealous type. Hera had already tried to kill Alcmene and Heracles by interfering with Heracles's birth, so fearing further nastiness from Hera, Alcmene abandoned Heracles in a field. (What? No mountain?) Luckily for Heracles, Athena came along and saved him by tricking Hera into breastfeeding him and eventually dropping him back off with his mom.