Many are called, but few are chosen. The phrase "world's greatest" truly applies on far fewer occasions than the number of times it's handed out.
The Rolling Stones started calling themselves the World's Greatest Rock and Roll Band at some point in the late 60s. It may not apply to their 2006 incarnation, but as long as their classic recordings continue to exist, it will apply in any moment you play 'em. It ain't bragging if you can back it up, and there are tracks in the Stones' library that aren't merely good or great, but that border on the miraculous. These records feature Keith's guitar drawing blood like a whip (I'm thinking of the intro to "Honky Tonk Women" or the first couple of seconds of "Start Me Up" and "Brown Sugar"), or Mick's voice dancing on the line between menace and decadence (as on "Bitch" and "Sympathy for the Devil"), or the whole band locking into a supernatural sync that's the very definition of rock and roll ("Jumpin' Jack Flash," "You Can't Always Get What You Want," "Gimme Shelter").
Any concern that stays in business 40 years is probably going to have its weak moments, artistic endeavors especially. And the Stones have certainly sounded flaccid or phoned-in now and then--most often, it seems, when they're playing live. If you check their discography at Allmusic.com, take note of the star ratings of their live albums: mostly in the two-star range, except for Get Yer Ya-Yas Out from 1970, but even though it gets 4 1/2 stars out of five, the review by Richie Unterberger says its appeal has dimmed today.
Now it's possible that they may never have been captured effectively on stage. But with at least 10 live albums in their catalog and few getting much more than faint praise, I doubt it. In fact, there's ample evidence to claim that The World's Greatest Rock and Roll Band is in fact The World's Greatest Studio Group. They're just not all that good playing live, and yesterday's Super Bowl halftime show was the latest evidence of it.
Granted, it has to be extremely difficult to play "Start Me Up" and "Satisfaction" anymore. One is almost 25 years old and the other's over 40, and familiarity has to breed just a little bit of contempt. But it's more than that. When they're live, Mick's singing seems to lose focus--it's as if he's concentrating more on firing up the audience and working the strut than on getting the words across. Keith's guitar retains some of its sonic menace, but with volume and distortion instead of the razor's edge he achieves in the studio. It's still rock and roll, but instead of blowing me away, it makes me conscious of how much better they sound elsewhere. As a result, although I've never seen the Stones live in person, it's not a priority for me to do so. I'd rather play Sticky Fingers again.
(Side note: I own Stripped, the Stones' live album from 1995, released when the "unplugged" craze was at its height. I got it precisely because I wanted to hear how they'd sound in more intimate settings, without the larger-than-life arena trappings of their other live albums. It's become one of my favorite Stones albums, with great versions of "Dead Flowers," "Sweet Virginia," and "Shine a Light," plus the inevitable cover of Bob Dylan's "Like a Rolling Stone." That might make it the exception that proves the rule.)
It came out today that the Stones consented to the electronic removal of potentially offensive words from "Start Me Up" and "Rough Justice." For people who remember the famous "Let's Spend the Night Together" brouhaha from The Ed Sullivan Show, it's painful to note that the Stones agreed quietly to comply this time. Maybe, with all those bluenoses sitting at home waiting for reasons to rain down their wrath on those they define as insufficiently moral, the Stones decided it would be easier to switch than fight. But then again, if anybody in rock and roll has the clout to tell the NFL and the Bible-beaters to go screw themselves, it would be the Rolling Stones, wouldn't it?