We don't talk a lot about the aftertaste, but if it's lacking, an otherwise enjoyable espresso is irretrievably marred. For me, the signs of overextraction are as noticeable in the aftertaste (bitter, metallic) as in the foretaste (harsh, missing sweetness, unbalanced). I've never smoked, but from what smokers describe, I think it's a similar pleasure they seek when then smoke after eating.
I don't recall an espresso with a fantastic aftertaste that wasn't at least a very good espresso overall. More to your point, I don't recall a great espresso that didn't have at least a good aftertaste. Curious, looking at the SCAA sensory judging sheet:
Color of crema (hazelnut, dark brown, reddish reflection)
Consistency and persistence of crema
Taste balance (harmonious balance of sweet/acidic/bitter)
Tactile balance (full bodied, round, smooth)
It's not listed. I'm not sure it fits cleanly in one of the categories of sensory perception defined above. It's part of the taste balance, but involves an "elusive it" I'm struggling to describe...