Saatchi Gallery, London: Rolling Stones' filthy west London flat on display at Exhibitionism

Keith Richards remembers it as a "pigsty". Mick Jagger reckons it wasn't actually that bad, it's "what we did with it that was disgusting". The problem, says Charlie Watts, was that "they were the laziest buggers in the world. They would never pick anything up."

The trio are referring to the one-bedroom flat that the Rolling Stones shared in Edith Grove, west London, when the band first started out. An immersive recreation of this bachelor pad, conjured from the Stones' collective memory bank, reveals a kitchen crawling with filthy plates, cutlery and fat-congealed pans; a bedroom and lounge littered with beer bottles, cigarette butts and discarded biscuits, and a stale, musty aroma that even the Stones, great merchandising machine though they are, (surely) couldn't bottle up and sell as perfume.

Yet despite – or, indeed because of – these almost comical levels of grottiness, most spectators are smiling or chuckling as they pass by. As Etta James' I Just Want to Make Love to You drifts from the lounge's antique vinyl player, the woman beside me says: "They've done a really good job of making it proper scummy."

Five and a bit decades on, with multiple platinum albums, 1800-plus live shows, untold riches and a place in popular culture assured, the Stones can afford to look back and laugh, too. Their "flat" is one of the most entertaining parts of Exhibitionism. Set in the stylish Saatchi Gallery, just off the King's Road, about a mile from their former abode, it's the first major exhibition to celebrate this most enduring of rock'n'roll outfits, whose seeds were planted when Jagger and Richards – who went to the same junior school – bonded over a shared love of blues music after a chance meeting at Dartford train station in 1961.

But Exhibitionism is no nostalgic valedictory. In the first of nine thematic galleries – in which glowing, high-tech screens chart the band's epic discography and marathon global tours – there's a quote from Richards: "It's still too early for me to talk about the Stones' legacy. We haven't finished yet."

This drive and ambition fuels a retrospective that showcases not just the band's prolific musical output, but their collaborations with filmmakers, photographers, visual artists, style and fashion designers, who've helped cement the Stones' fame and fortune. Jeff Koons, David Bailey and Martin Scorsese are among high-profile contributors, while one gallery highlights John Pasche's​ tongue and lips logo, which Jagger apparently commissioned after seeing a picture of Kali, the tongue-flaunting Hindu goddess, in his local Indian corner shop.

Pasche, then an art student, concedes his work may have unconsciously been based on Jagger's lips. Various incarnations of the logo – including one stuck on a giant cartoon gorilla – are among the 500-plus artefacts dotted around Exhibitionism.