I usually go for the up-tempo, swingier numbers (I'm shallow that way), so be warned that the slower, sadder songs will be under-represented on my list. But this one's a keeper: "I Could Have Told You"
Words by Carl Sigman (of "Marshmallow World" fame), music by Jimmy Van Heusen. Arranged by Nelson Riddle and recorded as a single in 1953. It was later added to the No One Cares CD.
Bonus: Here's a fabulous, jazzed-up version by the unlikely but wonderful duo of Eva Cassidy and Chuck Brown.
*Update: I see that Mark Steyn's audio tribute to Carl Sigman is once again available. I hadn't heard it since 2008 or so and had forgotten how entertaining it is. Quick, go burn a hard copy while you still can!