According to the Homeric Hymn to Aphrodite, when Eos asked Zeus to make Tithonus immortal, she forgot to ask for eternal youth (218-38). Tithonus indeed lived forever.
"but when loathsome old age pressed full upon him, and he could not move nor lift his limbs, this seemed to her in her heart the best counsel: she laid him in a room and put to the shining doors. There he babbles endlessly, and no more has strength at all, such as once he had in his supple limbs." (Homeric Hymn to Aphrodite)