The immortal character tends to drone off into long, winding speeches about how much better things were during the Hellenic Era. The characters listening to him are obviously bored, and have probably heard the same speech a dozen times before.
Zig Zagged: ???
Lampshaded: The immortal may say something like "What, did you think I'd keep something proving I'm a thousand years old lying around where anyone can find it?"
Played For Laughs: Declarations of immortality are met with flat disbelief or a character who is depicted as very old, but never directly referred to as immortal, does something that suggests they are. Like claiming your place of birth is a prehistoric continent.