In a room devoted to the Rolling Stones’ humble beginnings, there's a hand-typed Q&A from their fan club that asks Mick Jagger to state his personal ambition. His answer? "To own a business." That was in 1963. Well, he certainly got his wish. And this exhibition, the first major retrospective devoted to the band he's fronted for the past 54 years, shows how hard he's worked his rocks off to make it a success.

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It opens, rather strangely, with a re-creation of the flat that members of the band shared in Edith Grove, Chelsea, back in 1962. One suspects that the carefully art-directed squalor (a fag butt here, a casually discarded custard cream there) might hold some exotic appeal to far-flung visitors when the exhibition begins its global tour in September, but for anyone who has shared a student house in the UK in the past 50 years, it all looks depressingly familiar. Thankfully, that's the only misstep on your tour through the decades, which neatly dispenses with standard chronology to mix up the old and the new, neatly side-stepping that glaring quality drop-off around the mid-70s.

Spread across nine rooms and two floors of London's Saatchi Gallery, Exhibitionism explores the Stones' career through 500 objects, covering everything from personal journals and stage outfits to instruments and original artwork. For all the stuff that doesn't fit easily into the regimented areas covering "The Studio", "Film", “Design” and so on, there's a room of odds 'n' sods. If Charlie's kitchen sink was in there, you probably wouldn't be surprised, given the amount of material amassed here.

An entire room devoted to the band's working methods in the studio initially seems a rather dry experience – a place where music bores can geek-out over Keith's vintage Telecasters – but it's saved by a bank of virtual mixing desks that allow you to play producer with eight of the band's songs. I always thought the bass in Start Me Up was a little muted, so it's good to finally put that right.

One thing that becomes increasingly obvious as you navigate the Saatchi's rather haphazard floorplan is how much the Stones have collaborated. Their songs may be stuck in a chugging retro groove, but the band has regularly sought out the cultural cutting-edge – from Andy Warhol and Jeff Koons to David Bailey and Alexander McQueen – to present themselves to the world.

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Fashion has always played a big part in this business, and a hall of mannequins finds Brian Jones's dogtooth jacket and Keith's animal prints rubbing shoulders with Jagger's skimpy jumpsuits and the designer creations he wears on stage for Sympathy for the Devil. The partnerships don't end there, either, as the cavernous gift shop offers Hilfiger-branded T-shirts (at £40 a pop) and lips-logo Pringle jumpers (don't even ask). Men of wealth and taste, indeed.

But you've got to hand it to them, all those years as a touring band (1,823 shows and counting) means the Stones know how to put on a good show. The video installations help to break up all that incessant peering into glass cabinets, as does Martin Scorsese's short film about the various Stones documentaries down the years (including rare footage from the long-suppressed Cocksucker Blues). And the area devoted to album design is brimming with trivia nuggets, including the creative solution to the Sticky Fingers zip conundrum.

The final showpiece, a “3D experience” of seeing the band in concert, is cheesy as hell, but guaranteed to have you leaving with a smile on your face. It may also have you booking to see them play at your local enormo-dome soon. And that, Mr Jagger, makes for good business.

Exhibitionism runs at London's Saatchi Gallery until September 2016 before touring the world


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