Some that haven't been mentioned but fit the category in one way or another are:
WHEN I FIRST CAME TO THIS LAND
SUNRISE, SUNSET
TURN AROUND
I'M GLAD I'M NOT YOUNG ANYMORE (Sung by Maurice Chevalier)
YOUNG AT HEART
Angel from Montgomery (John Prine)
Growing Older But Not Up (Jimmy Buffet)
A DAISY A DAY
KISSES SWEETER THAN WINE
Hello, Young Lovers
THE OLD GRAY MARE, She Ain't What She Used To Be
Old Hippie (The Bellamy Brothers)
Believe Me, If All Those Endearing Young Charms
OLD OAKEN BUCKET
Love, Me (Colin Raye)
I'LL TAKE YOU HOME AGAIN, KATHLEEN
Amanda (Don Williams)
Madame Jeannette

And there's a song I heard in a movie that was set, I think, in the 19th C., and had the line,
"I am old and gray and I've lost my way, All my tomorrows were yesterday ..."
Can't recall the rest of it. (I think a very young Angela Lansbury was singing it, and it may have been in the movie "Bel Ami.")

How about Edith Piaf's theme song, "Non, Je Ne Regrette Rien?"
She also sang a great song about "Le Vieux Piano" ("The Old Piano") which has just died. It's a metaphor for the aging of the folks who played it and heard its music.