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For one single "Greatest Hits" type compilation I recommend "Classic Sinatra: His Greatest Performances 1953-1960," which came out a couple years or so ago. Should be easy to find (online I like Amazon and Tower Records). It has many of his greatest tracks including a couple of the songs mentioned in posts above.

The thing you do miss out on with a "Hits" compilation, though, is that starting in the '50s his albums were designed with each song chosen to fit a certain theme or mood. You can't go wrong with most of his '50s-'60s albums for Capitol and Reprise. For a single title to get you started I'd go with the upbeat "Songs for Swingin' Lovers!" (This is the one that's my teenager's favorite.)

If you start browsing Sinatra on Amazon you should turn up a couple really good lists by Amazon customers reviewing some of Sinatra's other "theme" albums.

Hope you'll enjoy! :)