1. The Default Option: The Funeral Games. Paris will be ~16 here. Upon having his favourite and most well-bred bull (he might or might not have let one of the royal bulls cover one of his foster parents’ few own cows ahem) taken as a prize for the annual funeral games held for a dead Trojan prince, and not wanting to give it up, follows the men down to Troy despite Agelaos’ attempts at dissuading him. He wants his bull back, especially when it doesn’t actually belong to the royal herds. He argues against the king to be allowed to participate in the athletic games held, and manages to win enough of the individual contests he’s declared the victor of the games entirely. This insults the princes, and Deiphobos, after arguing with Hektor and then convincing Hecuba, attempts to murder what they think is nothing but an “upstart” slave. Paris flees to the courtyard/entrance altar of Zeus Herkeios to take refuge. During this his (foster) father appears and shows the royals the medallion Paris has been in possession of since an infant, which has his name scratched into the gold on the back (Priam himself did that, the medallion was his).
(Sources: The funeral games + murder plot/attempt by Deiphobus is earliest attested in Sophocles and Euripides' fragmentary/mostly lost Alexandros play, of which a substantive hypothesis survives. Here is Hyginus' (Fabula 91) summary/version, which accords well/basically exactly with the hypothesis: When he came to young manhood, he had a favorite bull. Servants sent by Priam to bring a bull to be given as prize in funeral games in Paris' honor, came and started to lead off the bull of Paris. He followed them and asked them where they were leading him. They stated that they were taking him to Priam . . . [to be prize] for the victor in the funeral games of Alexander. He, out of fondness for the bull, went down and won everything, even over his own brothers. In anger Deiphobus drew his sword against him, but he leaped to the altar of Zeus Herceus. When Cassandra prophetically declared he was her brother, Priam acknowledged him and received him into the palace. (It's of course more complicated and involved than this, and Hyginus doesn't mention Agelaos/Paris' adoptive father coming to the rescue, Hektor not being on board doing anything and potentially protecting Paris, or Kassandra telling everyone to kill him with her reveal of who he is/the prophecy/dream omen he was born with.))
2. The Younger Option: Here Paris is found (by one means or another, default assumption when not playing against a Hector will be that Hector found him while hunting, but Aeneas or Anchises is another option, given that they have herds on Mount Ida) at ten and brought back to Troy. A slightly less dramatic and potentially lethal reveal situation, with a couple more extra years for Paris to be with his birth family.
From early teenage years he slowly spends more and more of his time on Mount Ida herding Troy's royal cattle, as this is a perfectly fine occupation for princes (also due to lingering parental worries and guilt). This leads both to a relationship (though not marriage) with the nymph Oenone for a little while shortly before he goes to Sparta and the chance to pit his pet/prize bull against others and, finally, against a disguised Ares, as he'd promised a golden crown as a prize if his own bull could be defeated.
3. Growing Up in Troy: As briefly touched on above, in the Iliad there is no mention at all of any exposure, or even of the dream omen. This doesn't mean neither of these things weren't known yet, but the Iliad, despite opportunities, makes no mention of them. So either they might have existed, yet not accorded in the text, or they hadn't formed yet, which means in the Iliad's version of things Paris grew up from infancy with his family, which gives yet another possibility of interactions between Paris and his family. (I can take or leave the dream omen here. If included, maybe Hecuba and Priam simply couldn't give him away to be killed, or, say, Hector found out what was going to happen and pleaded for Paris to stay.)
Background variations
The Funeral Games. Paris will be ~16 here. Upon having his favourite and most well-bred bull (he might or might not have let one of the royal bulls cover one of his foster parents’ few own cows ahem) taken as a prize for the annual funeral games held for a dead Trojan prince, and not wanting to give it up, follows the men down to Troy despite Agelaos’ attempts at dissuading him. He wants his bull back, especially when it doesn’t actually belong to the royal herds.
He argues against the king to be allowed to participate in the athletic games held, and manages to win enough of the individual contests he’s declared the victor of the games entirely.
This insults the princes, and Deiphobos, after arguing with Hektor and then convincing Hecuba, attempts to murder what they think is nothing but an “upstart” slave. Paris flees to the courtyard/entrance altar of Zeus Herkeios to take refuge. During this his (foster) father appears and shows the royals the medallion Paris has been in possession of since an infant, which has his name scratched into the gold on the back (Priam himself did that, the medallion was his).
(Sources: The funeral games + murder plot/attempt by Deiphobus is earliest attested in Sophocles and Euripides' fragmentary/mostly lost Alexandros play, of which a substantive hypothesis survives.
Here is Hyginus' (Fabula 91) summary/version, which accords well/basically exactly with the hypothesis:
When he came to young manhood, he had a favorite bull. Servants sent by Priam to bring a bull to be given as prize in funeral games in Paris' honor, came and started to lead off the bull of Paris. He followed them and asked them where they were leading him. They stated that they were taking him to Priam . . . [to be prize] for the victor in the funeral games of Alexander. He, out of fondness for the bull, went down and won everything, even over his own brothers. In anger Deiphobus drew his sword against him, but he leaped to the altar of Zeus Herceus. When Cassandra prophetically declared he was her brother, Priam acknowledged him and received him into the palace.
(It's of course more complicated and involved than this, and Hyginus doesn't mention Agelaos/Paris' adoptive father coming to the rescue, Hektor not being on board doing anything and potentially protecting Paris, or Kassandra telling everyone to kill him with her reveal of who he is/the prophecy/dream omen he was born with.))
2. The Younger Option:
Here Paris is found (by one means or another, default assumption when not playing against a Hector will be that Hector found him while hunting, but Aeneas or Anchises is another option, given that they have herds on Mount Ida) at ten and brought back to Troy. A slightly less dramatic and potentially lethal reveal situation, with a couple more extra years for Paris to be with his birth family.
From early teenage years he slowly spends more and more of his time on Mount Ida herding Troy's royal cattle, as this is a perfectly fine occupation for princes (also due to lingering parental worries and guilt). This leads both to a relationship (though not marriage) with the nymph Oenone for a little while shortly before he goes to Sparta and the chance to pit his pet/prize bull against others and, finally, against a disguised Ares, as he'd promised a golden crown as a prize if his own bull could be defeated.
3. Growing Up in Troy:
As briefly touched on above, in the Iliad there is no mention at all of any exposure, or even of the dream omen. This doesn't mean neither of these things weren't known yet, but the Iliad, despite opportunities, makes no mention of them. So either they might have existed, yet not accorded in the text, or they hadn't formed yet, which means in the Iliad's version of things Paris grew up from infancy with his family, which gives yet another possibility of interactions between Paris and his family.
(I can take or leave the dream omen here. If included, maybe Hecuba and Priam simply couldn't give him away to be killed, or, say, Hector found out what was going to happen and pleaded for Paris to stay.)