I've been back-burner thinking about this off and on, and I think, for me, I would nominate a band for at least recognition in the category. A band mentioned before in this thread - Radiohead.
For me, the progression in their work is impressive. Their desire to change and go places musically. I remember first hearing them, and thinking they were a talented but perhaps somewhat 'of the era' grunge-type band. A cut above it's true but none-the-less a little bit playing to a crowd. 'Creep' gave them a following, a popularity, that only really associated them with other acts, confirmed their place in a teen-angst crunge fold. But then, rather than cement that, they moved on. Their 'OK Computer' album is nothing short of stellar and has to be considered amongst the best albums of the decade, if not 20th century. Then they moved on again, away from standard rock beats and pursuing a different musical direction and playing around rhythmically.
As for influential, you'll hear their work (either as a band or as individual musicians) in the soundtracks of many a film or TV show. And, of course, many other bands and musicians have taken strands from their work as the basis for their own work.
Now I know they're not to everyone's taste (misery music is an oft heard criticism!) but for me they are certainly up amongst the 'best' bands.