It's not the vertiginous heels that I can't keep my eyes off, nor the super-mafioso shades, not even the see-through catsuit. It's that little growth from her forehead. There's only one on show today. It could be a horn, a cancer, an embryonic phallus. It could be, as she likes us to think, a physical manifestation of her creative genius. And it could, of course, be a prosthetic implant as fashion accessory.
She's chatting away about how touring is her life, how she loves her fans like no other artist, her heroes, and all I'm thinking about is the growth.
"Can I touch it?"
She smiles, and doesn't answer.
"What is it?"
"It's pleased to see you. They come out when they're excited."
We're sitting in her "relaxation" room at the Q arena in Cleveland, Ohio. There are numerous perfumed candles, a supersonic hi-fi system - all the normal rock 'n' roll paraphernalia. On the walls are photos of Led Zeppelin, Iggy Pop, the Sex Pistols. Gaga is a musical magpie. She's brazenly nicked and nicked and nicked to create something her own. There's no attempt to hide it, either. So on her first album, The Fame, which pre-empted her actual fame, she thanked Madonna and Bowie and Prince.
At a glance, you can see the bras and knickers of Madonna, the hats of Grace Jones, the body suits of Bowie. We're listening to bits of her new album, and you can hear Giorgio Moroder, the E Street Band, Euro pop. Whereas her first album instructed her fans to embrace their inner fame, this one could be addressed to a younger audience - follow your heart, no matter what your parents say. It's called Born This Way, after the single that was number one for six weeks in the United States.
The irony, of course, is that Gaga was not "born this way" but, as she says, to interpret it literally is to miss the point - we're born this way to be true to ourselves, whatever that is.
She explains the song Hair. "Hair is about when you're younger. I am my hair." And was it really her hair, or was she cheating it back then? "No, I wasn't cheating it back then."
She looks at the coloured Rapunzel locks flowing from her hat. "I wouldn't say I'm cheating it now. I would just say it's a surrealist extension of my being. I'm half living my life between reality and fantasy at all times. It's best not to ask questions and just enjoy."
Which is a perfect Gaga statement. If you're not wedded to reality or truth, you're entitled to say whatever you fancy. On the wall is a picture of Warhol's painting of Debbie Harry. Warhol is perhaps Gaga's biggest influence. Just as he was, she is obsessed with the nature of fame and the marriage of commercialism and fine art.
Gaga is just about to perform her last US date of a world tour, The Monster Ball. When she started it in November 2009, the performance artist from New York was beginning to enjoy an unlikely success. Now she is the biggest music star in the world. Nobody is quite sure how she did it, least of all her. What's more, she doesn't seem to have gained fans so much as followers.
It's no coincidence that her latest single is called Judas, and uses that great biblical betrayal to tell a story of love gone wrong.
After singing about fame on her first two albums, she says she's bored with the subject. "On Born This Way, I'm writing more about pop culture as religion, my identity as my religion: 'I will fight and bleed to the death for my identity.' I am my own sanctuary and I can be reborn as many times as I choose throughout my life." She has never shied away from the grand statement.
She calls her fans little monsters and they call her Mother Monster. Already, she says, she has undergone so many transformations - from Stefani Joanne Angelina Germanotta to Gaga to Mother Monster.
"I bet your mum doesn't call you Mother Monster," I say. "No. She's the only mother in our house. She'll just call me Gaga or Stefani." "How long has your mum called you Gaga?"
"I started being called Gaga by friends when I was about 19, and I don't believe she stuck on to it right away, because she was a bit worried about my mental health." She bursts out laughing.
Outside the arena in Cleveland, the fans are gathering - young men and women, girls and boys. Support act Semi Precious Weapons, a glam rock group, are getting ready. Gaga's first gig in 2007 was supporting them in front of 400 people in New York.
Today, a Gaga event is ridiculously lavish and expensive, what with the meat dresses, the dance troupe, the many designers, and the Haus of Gaga entourage. A recent photo shoot was estimated to have cost £150,000 ($307,600).
Gaga's show starts with a giant projection of her. Then the curtain goes up, and there she is in all her lunatic majesty. This is not a gig, it's theatre. The set is a degenerate hell, with neon directions to Liquor, Implants, Sedation, Car Accident, Death Cases and Whip Lash. There are crashed cars, voracious monsters, trees with razor-blade leaves. It's hard to follow the plot, but she seems to be passing through the valley of self-harm.
The outfits change by the song. One minute she's all formal elegance, the next there are streaks of fire shooting from her bosom and crotch. She jumps on the piano and plays with her feet. She douses herself in blood - cheeks, shirt, belly, all haemorrhage red.
"Do you think I'm sexy?" she shouts. "It took me a long time to feel sexy because I was bullied at school." She is an astonishing performer, and she knows it. "In case you're wondering whether I lip synch, the answer is no ... people think so because I sound so good."
As she sings the hits - Paparazzi, Poker Face, Just Dance, Bad Romance - she tells her disciples:
"I didn't used to be brave, but you have made me brave, little monsters." Gaga is an evangelist. "Nothing we do together has a last or first time," she says, her voice cracking with emotion. "We are eternity ... you made me, little monsters. Tonight I want you to forget all your insecurities ... what you feel makes you different in the greater destiny of life ... I worked so hard to get where I am ... just remember, I was so far beneath and now I'm so far above."
Perhaps she's simply playing a role - she knows the iconoclasm, the Jesus parallels and parables, are all of a piece with flogging her new single, Judas. But I think there's more to it than this. She is a sincere believer in the cult of Gaga - she really does think she's a modern-day Messiah, here to lead her fans to a brighter, better future. And if they buy the records and spread the word along the way, so much the better. Gaga has now set her sights on breaking into India - a vast, untapped market.
Stefani Germanotta was born in New York to self-made parents - her father, Joseph, was an internet entrepreneur, mother Cynthia a telecommunications executive. She started playing piano at four, attended a private convent school on Manhattan's salubrious Upper East Side, escaped to the less salubrious Lower East Side to perform at open-mic nights at 14, won a place at New York University's Tisch School of The Arts at 17, where she studied music and left before completing her degree because she was miserable and impatient for the real thing.
At 19, she signed to Def Jam and was dropped in three months. A couple of years later, she was writing songs for New Kids On The Block and Pussycat Dolls. By 2008, she was living in Los Angeles recording an album that's sold 14 million copies.
A day after playing Cleveland, the Gaga-mobile is on the road to Chicago where she will sing on one of Oprah Winfrey's final shows. The Haus of Gaga, the name she has given to her creative team, has taken over the top floor of a luxury hotel and transformed it into a bustling factory. In one room, her video diary is being updated, in another the Oprah set is being designed, and in another outfits are being vetted. Gaga, now 25, strides from one room to the other, huge and imperious in wraparound shades, mega heels and black Alexander McQueen powersuit. It's easy to forget she's only 1.58m tall.
I am taken to a room to wait for her. One is always placed in situ so she can make an entrance. Today, she marches in and apologises for all the shouting in the background - things haven't been going too well. She takes off the shades and softens - her skin is gothic white, her face unremarkable. Her jacket is barely buttoned, and she is wearing nothing underneath. It seems like a statement outfit - "look at this and try telling me I'm a man (as has been rumoured) or anorexic (ditto)".
The previous day she had said it's difficult to talk about the show to people who haven't seen it. She was right, I say, it's like a revivalist meeting. "Yes, it is. It is a religious experience. But it's like a pop cultural church." She pauses. "I never intended for the Monster Ball to be a religious experience, it just became one."
She's become the Billy Graham of pop, I say. She laughs. "It's more self-worship, I think, not of me. I'm teaching people to worship themselves."
What is going wrong with conventional religion if kids are looking to her for spiritual guidance? "The influence of institutionalised religion on government is vast. So religion then begins to affect social values and that, in turn, affects self-esteem, bullying in school, teen suicides, all those things." Her message, she says, is simple and perfectly Christ-like - love yourself and love others.
Does having such influence scare her? "No, but it does put me in an interesting position as an artist whose fan base is commercial and widening. If you were to ask me what I want to do, I don't want to be a celebrity, I want to make a difference. I never wanted to look pretty on stage and sing about something we've all heard about before. I'd much rather write a song called Judas and talk about betrayal and forgiveness and feeling misunderstood, and talk to the fans and figure out what it is society needs. If I can be a leader, I will."
Where did her own sense of being misunderstood come from? "From being bullied." Wasn't she too tough to bully? "No, I was not hard. I'm eccentric and talkative and audacious and theatrical, and I used to get picked on. I got thrown in a trash can on a street corner once by some boys who were hanging out with girls in my class. I got profanity written all over my locker at school and all the others were nice and clean. I got pinched in the hallways and called a slut."
What did she feel like when they put her in the trash can? "Worthless. Embarrassed. Mortified. I was 14. Three boys put me in it. The girls were laughing when they did it."
There have been other struggles - drugs in her late teens, discovering she carried the lupus gene from which her aunt died, being dropped by Def Jam. Could she have become who she is without having gone through all this? "No, I don't think anybody could. That is, in many ways, the point, and the point of Judas; in fact, Judas was not a betrayer, he was just part of the over-arching destiny of the prophecy. Those things in your life that haunt you are just part of what you must go through in order to become great."
It's hard to know how to respond. The strange thing about Gaga is that for all her outlandish outfits and statements, I find it hard to remember what she looks or sounds like the next day.
On her left wrist is a peace symbol tattoo. In her dressing room, there is a picture of John Lennon and Yoko Ono in bed holding up the slogans "Hair peace" and "Bed peace". As she talks, I think of Lennon's statement, "we're more popular than Jesus now". Just like Lennon, she insists she's not anti-religious.
"Don't say I hate institutionalised religion. What I'm saying is that perhaps there is a way of opening more doors, rather than closing so many."
Your mix of indomitable self-belief and uncertainty is weird, I say. "I always had a lot of nerve. And insecurity." The two go side by side? "Why not? I don't believe any of the most exciting artists in history were these privileged, happy zen human beings. The struggle of so many artists is why I fell in love with them." Such as who?
"We all know how much I love David Bowie." Not only has she adopted the royal "we", she is now speaking in a voice resembling that of the English upper classes. "We know how much I love Elton John." Gaga is godmother of Elton and David Furnish's son, Zac.
Is it true that as a child she prayed for madness? "Yes, I did, and here we are," she says with a little mousy smile. "Careful what you wish for." What was the madness she craved? "I wanted to release my true identity, which I had not yet fully discovered. I used to stare at all the posters on my wall, many of which are the same ones that are on the walls of my dressing room now. I used to imagine what these legends must have been like and then I used to pray that whatever condition was impressed upon them I would also experience. In a lot of ways it was a prayer for creativity."
Looking at those posters in her dressing room, I was struck by the similarity of their stage presence - most of them are topless, Iggy Pop virtually naked. It made me think Gaga's own state of undress was not so much a sexual provocation as a statement about being a rock star. "Absolutely. It is a statement about show business. I adore show business and don't ever want my fans to see me in any other way. I don't believe in it. Show business, pop culture, that is my religion."
I tell her that others would not have been able to cope with Gaga's fame. "I have no conception of how or how not famous I am. I have no conception about how successful I am. I can only conceive of what happens at the Monster Ball."
But a couple of minutes later I discover she's not always quite so humble. We're talking about Born This Way, and I momentarily forget its name. She looks at me in disbelief, then at her publicist, horrified. I think she's joking, but there isn't the flicker of a smile. "It's really great that you're writing about my music and you don't know the name of my record. Is it forgettable?"
"No," I say. "Look, I can even sing it for you." So I do, complete with actions.
"That was good." The smile returns. "I like the fist pump."
Would it upset her if I couldn't remember the name of the album? "Yes, it would. And it would upset all my fans."
I tell her that loads of people think she is the kind of person who would climb over her friends and claw her way to the top. "I would never do that. There's nothing that makes me happier than to see my friends' success. Money doesn't matter to them or to me. What matters is that we did it."
She's welling up. "What matters is there was a time - I'm going to cry - when nobody knew who we were, and all we had was each other and we only created for each other, because it made us happy, and now that the whole world is watching, we're still doing the same thing. So, no, I didn't pop out of nowhere and become a pop singer. It's been a very real and arduous journey through Judas all the way to Jesus." Just as she's talking from her heart, she has to go and ruin it all with the Judas-Jesus shtick.
Yes, she says, she knows there are misconceptions about her, but she doesn't care. Such as? "If you are eccentric or have a strong identity or live in a fantastical regalious way ..."
"Regalious? I don't know. I made it up. There is a belief that if you're strong in your identity or different, there must be nothing substantial underneath. It's hiding something vacuous or it's artificial. Whereas I am solidly devoted to my artistry and craft, but am also quite regalious - whatever it means."
Will she ever have enough, and retire to a nice, quiet world of non-Gagadom?
"I don't separate myself from Gagadom. So, no ... I do think I would like, maybe next week, to retire to somewhere quiet, and still exist in Gagadom. That's who I am."
In recent months there have been hints of a backlash. Perhaps it's inevitable. There are critics from gay and lesbian organisations who say she is not gay or bisexual enough (she has said she's had relationships mainly with men, occasionally with women); religious leaders who say she is blasphemous; former producer Rob Fusari who says he invented her; school peers who say she wasn't bullied.
"That's my favourite. 'You weren't bullied.' You're bullying me by saying that. Hahahaha! Isn't that a funny statement, 'You weren't bullied'?"
Can she brush it off?
"What I have learned the most from my fans, and what you will hear most on Born This Way, is that it is part of my destiny to take the bullets, but my heart keeps on beating. You cannot destroy me. Because I am an art piece." And with that, she's off.
Next day at the Oprah show, the curtain comes up to reveal a giant golden stiletto with Gaga perched at the top playing piano, singing Born This Way. It's a wonderful performance - slowed down, anguished and a sign that she could evolve into a brilliant soul or jazz singer.
Three days on the road with the Gaga industrial complex, and you see her in all her contradictory glory. So there she is, ruthless marketeer and hippie chick, needy neurotic and self-proclaimed saviour, younger than her years and older than her years, and ultimately rather brilliant - though it would be impossible to be quite as brilliant as she'd have us believe. After Oprah, she meets her host backstage. She's wearing pants and a bizarre face mask, and looks like a lost little girl. She pats her tummy and does a self-conscious little dance.
"Oooh, Johnny Depp," she says in awe as the actor passes.
"Lady Gaga!" he says. For a second she can't believe he knows who she is. And then the confidence returns.
"Was I good?" she asks one of her team.
"You were good," he says.
"You were good."
"Good. You were good."
"You were amazing," says another of the entourage.
"I prefer you to him," she says, and already she's talking about how her tour will be remembered for ever. She says most people measure wealth in terms of record sales, but not her.
"Success is not temporal. You either have it for ever or you don't. My wealth is in the cultural impact I have and in my longevity and legacy and the stamp that I will leave on the universe."
What will that be?
"She was always herself. She was fearless, and she moved pop culture, technology, art and music forward relentlessly, and she never gave a f*** what anybody thought about her."
Trust Gaga to have already written her own obituary.
Born This Way (Universal Music) is out now.