Originally Posted by AuAAdvantage
Actually, I was more concerned about being delayed by them and
missing the "next plane back to the U.S.", i.e. mine.
If they "reject" you the airline that delivered you gets the honor of returning you. You'll make it home one way or another. That's why the airline also checks your passport before you leave.
As long as your passport is valid and you still bear a vague resemblence to your passport photo, you won't have anything to worry about.
Depending on how the airport is arranged and whether or not a transfer desk is available to provide your boarding pass for the return, it's conceivable that you might not even go through customs.