It’s the view, however, that marks it out as the home of a rock star. With 360 degrees across London, you can see the London Eye, the Gherkin, Westminster, Canary Wharf, Hampstead Heath and the South Downs. Or, if you’re Robbie Williams, you can see trouble.

“See that apartment over there?” he says, lighting a fag, sliding open the doors and moving out on to the balcony. “A woman does coke there every night. And see that apartment there? A really fat old ugly bloke walks around naked and never draws his curtains.” He crosses over to his second balcony and gestures. “He does coke” — gesturing to a window — “and she does coke. And down there — that’s where this black woman — nothing wrong with that — naked — nothing wrong with that — came out dancing on her balcony. She started waving to me. Then she started singing and I thought, I’m buggered if I’m waving back now.

“I can’t believe I left my night-vision binoculars in LA,” he says, slightly regretfully, as he steps back into the apartment and reaches for another cigarette. “Chain-smoke, shall I? Why not.”

It’s been another busy week for Robbie Williams, arguably the world’s biggest pop star. On Friday, he broke his arm. On Saturday, he officially launched his new album in “the heart of Europe” — a 7,000-capacity audience in the Velodrom, Berlin, complete with 12-piece orchestra and 12-piece choir. On Sunday, the chart-placing of new single Tripping was announced — No 2 in the UK, but Billboard’s No 1 single in Europe, and the airplay No 1 across South America. On Monday, it was announced in the tabloids that Take That would be re-forming for a one-off gig. Ask him about the rumour and Robbie is unequivocal.

“I can say something nice about every single member of the band,” he says, looking down at his hands as he smokes his cigarette. The nails are so bitten that the tips look like stumps. God knows how he peels an orange.

“But when it comes to Nigel Martin Smith (Take That’s former manager) . . . I want to rip his uterus out.”

Take That will not be reforming.

Having finally concluded his tour of the seedy side of life around Chelsea Harbour, we abandon the balcony. Williams sits on his big sofa and starts scratching his broken arm maniacally.

“I’ve broke my arm, by the way,” he says, holding it out, somewhat unnecessarily. A large purple bruise bisects his arm, in the middle of his Born To Be Mild tattoo. Playing football he fell — “a proper cartoon fall, like Norman Wisdom” — and sustained a fracture. The arm should be in plaster, but it was “too hot and uncomfortable”, so he took off the plaster. He habitually returns to pawing at it, like a cat after an operation.

We do, of course, have a lot to talk about this afternoon. He is temporarily breaking his self-imposed two-year exile in Los Angeles to promote his new album, Intensive Care, in the UK. Intensive Care is an album with a lot on its mind. Darker, tougher and more savage than its more vaudevillian predecessors, and infinitely more accomplished, the album sees him admit to an affair with a married woman, finally mourn the break-up of his relationship with a famous ex-girlfriend, compare himself to Elvis, wish himself back as a teenager in a cheap nightclub in Stoke, turn to black magic to conquer his fears and, in the most ear-catching moment of the album, starkly claim that “I’ve got a head full of f***/ I’m a basket-case/ I don’t know how to love, love, love”.

Given this pressing agenda of subjects to be discussed, then — matters that touch on the very core of Williams’s much-vaunted unhappiness and spiritual unease — it’s no surprise that he starts our interview with a six-minute rant about his feud with the camp LIVINGtv psychic Derek Acorah.

A fan of Acorah’s TV work, Williams summoned the psychic to the Chelsea penthouse for an audience, because that’s the kind of thing you can do if you’re Robbie Williams. However, when pressed by Williams to “go on and do it” — contact the dead — Acorah came up with little more than what a search would yield on Williams’s dead relatives, albeit rendered in a blaze of peerlessly camp Scouse dramatics.