The incredible musicianship is obvious. Even people who don't like their music will acknowledge that the playing itself is first rate. I also really love the lyrics and the little sonic world that each song inhabits. The songs are more like musical movies than just "singles."
What I really love about their lyrics is the way they embrace losing. Many, many of their songs are about losing, or losers, or falling short. In fact I'd say that the main subject of their writing is about the distance between what we want and what we get, as well as the joy, still be found, with what we do get.
If you think of them as just some ultra-slick studio boys who create a veneer of irony and riff on their own coolness...well, you're right. But they are also so, so much more. I think of them as true classic American Jazz artists. And amazingly, I think that literally any one of their many songs illustrates this. I can't think of a single song that I don't at least love. So I can't even recommend one song over another. I guess if I had to, I'd recommend their album "Aja," but the minute I even think that I regret not choosing "Katy Lied" or "Two Against Nature" or "The Nightfly" or "Gaucho" or really any of their albums.
PS: I'm adding this some weeks after first posting this. I've been thinking further about why I like this band so much and don't get tired of listening to them. I think it really comes down to the intelligence imbedded in every moment of their songs. Musically, emotionally, and intellectually, there is always a precise intelligence at work. And that works for me. I love other aspects of music and of people too, but inevitably it's intelligence that holds my interest best.