He is in the middle of shooting a movie – called, appropriately enough, The Surfer – though it would be a shock if he wasn’t. Impersonating Cage on Saturday Night Live, Andy Samberg said: “My dream as an actor is to appear in every film ever released.

However, until now I’ve only been able to muster a measly 90%, bringing shame upon my dojo.” At one point during our conversation, Cage claims not to know how many films he has made.

I count 109 completed features, though the ones in which he plays dual roles, such as Adaptation and The Unbearable Weight of Massive Talent, should probably be counted twice. For Unbearable Weight, where he played two Nicolas Cages, he even snogged himself. “I wanted it to be kind of cubist,” he says.

Some guy called my manager to ask if I was interested in buying a two-headed snake. I couldn’t say no

This year, he has already starred as Dracula in Renfield, an assassin turned beach bum in The Retirement Plan, a crimson-haired car-jacker in Sympathy for the Devil and a brooding buffalo hunter in Butcher’s Crossing.

(He also had a near-subliminal cameo as Superman in The Flash.) Dream Scenario, though, is his pick of the bunch. “I’ll go on record as saying I think it’s a masterpiece,” he says.

Cage plays Paul Matthews, a schlubby college professor who starts popping up in strangers’ dreams, making him a kind of accidental celebrity of the collective subconscious. It might be flattering if he didn’t ignore the dreamers’ cries for help as they are pursued by zombies and whatnot.

Though the actor claims not to set much store by his dreams, he has turned to them for help. “If I’m stressed about how to play a scene, I’ll ask for a gift from my dreams.”

Only five times, he says, has he been blessed with a script that didn’t need a single word changed: Dream Scenario, Adaptation, the Coen brothers’ madcap masterpiece Raising Arizona, the 1988 comedy Vampire’s Kiss, which set a high bar early on in his pursuit of sublime derangement, and the recent Pig, where he was movingly low-key as a former chef whose porcine BFF is snatched. “Sometimes I’m so nervous about not having any hook on a scene that I’ll go to bed and dream about it and then I’ll get some residual feeling that I can apply to the performance.”

‘I’ll go on record as saying I think it’s a masterpiece’ … Nicolas Cage in Dream Scenario.

His dreams have also influenced other choices. There were two-headed eagles, for example. “Isn’t that so wonderfully bizarre?” he gasps. “Yes, I was dreaming about two-headed eagles – I don’t know what to take from that – and then the next day, some guy called my manager to ask if I was interested in buying a two-headed snake.

I couldn’t say no because I was connecting the dots. It was too beautifully weird.” However, when he discovered the rigmarole involved – the heads had to be kept apart with a spatula to prevent them squabbling over dinner – he donated the snake to a zoo. “Crazy stuff.”

Dream Scenario rang some bells for him. “Everyone has the ability to become famous now. You’re the guy who tripped in the supermarket and someone filmed it. Or the woman on the plane who has a meltdown and now you’re the Airplane Meltdown Woman.”

In a funny way, he knows what that’s like. Yes, he already had two decades of fame behind him when the internet became commonplace: he started acting in the early 1980s, won an Oscar in 1996 for playing a bruised alcoholic reaching the bottom of the bottle and the end of the road in Leaving Las Vegas, then pivoted to action hero with the boisterous hat-trick of The Rock, Con Air and Face/Off. But celebrity at that level still left him unprepared for the meme scenario.

“I might have been the first actor who went through a kind of meme-ification,” he says, alluding to the proliferation of memes (such as “You Don’t Say!”) along with countless video variations on the theme of Nicolas Cage Losing His Shit.

“One person had cherrypicked from all these different movies where I was having meltdowns, but without any regard for how the character got to that place. I was frustrated because I didn’t know what people were taking from the movies other than that.”

Part of the reason Cage provides such rich pickings is that he has never knowingly been inhibited. No one else speaks in so many italics and block caps. Listen to the way he pronounces “testicle” during his crazed speech in Prisoners of the Ghostland (“TESTICAAALLL!”).

Watch him bad-mouthing a belligerent guard-dog in David Gordon Green’s Joe (“I love dogs! Just not that dog! That dog is an asshole!”) or going cuckoo on crack in Werner Herzog’s Bad Lieutenant: Port of Call New Orleans (“Easy-easy-easy-easy, ’cos I’m not Eazy-E!”). These doolally distillations of character are equivalent to an electrifying riff from a rock song: it’s not the whole number but it sure whets the appetite.

As a child, Cage had already seen fame up-close. His aunt, Talia Shire, was a star from The Godfather (directed by her brother – and Cage’s uncle – Francis Ford Coppola) and Rocky long before he had even made his first movie. “I’m so glad you’re bringing up aunt Tally,” he says warmly. “When I was only seven or eight, she was the one who said to me: ‘Naturalism is a style.’ That stuck with me.”

Being known for memes and mashups, though, is different. “I didn’t understand how to process what was happening. I got into acting because I was moved by film performance more than any other art form. I didn’t get into movies to become a meme.

That was new. I made friends with it but it was an adjustment.” He convinced himself that these were memes to an end. “I thought maybe they would compel someone to go back and look at the movies. But I had no control over it. The same thing happens with Paul in Dream Scenario: he has no control over this inexplicable phenomenon.”

Paul is also under the illusion that he can exploit his unsolicited fame and ubiquity. “Look, I don’t gamble but fame is like gambling,” says Cage. “When fame turns on you, the effects of loss, as with gambling, are more profound than the effects of winning.”

Has fame turned on him? “Oh yeah! There’ve been moments when things were broadcast about my finances, or mistakes I’d made became extremely public, and I felt all that.”

He is referring to his brush with financial ruin about a decade ago, when he blew his fortune on extravagances including two castles, an island in the Bahamas and one of the US’s most haunted houses, and emerged with a $14m bill for unpaid taxes.

(He later sued his business manager for making “numerous highly speculative and risky real estate investments”; the manager countersued, citing Cage’s “compulsive, self-destructive spending”. The suit was settled in 2012.)

Perhaps it’s true to say that he was hurt but not defeated – to echo a phrase that Martha Coolidge used when she directed him in Valley Girl back in 1983, as she tried to get him in the correct mindset when his character suffered a knockback. “Oh my God!” Cage exclaims.

“That was the greatest piece of direction I’ve ever received from a film-maker. I’m so glad you mentioned Martha because I want to go on record as saying that without her, none of this would have happened. It was because of what she did with me that people knew who I was.

That was so long ago. You know, I’m going to be 60 in January.” This he says with audible disbelief. I offer my congratulations, and he thanks me with such sincerity that it is almost as if he believes I went out and bought that “congratulations” specially for him in his size.

That’s Cage all over: a master of irony, sarcasm, rage and cruelty on screen but a fountain of warmth in conversation, determined to make the person he is talking to feel special.

Four or five times, for instance, he begins an answer with some variation on the words: “I’m so glad you’re bringing that up …” At the end of our time together, he even offers a kind of customer-satisfaction wrap-up: “That was a good conversation in my opinion and I hope it was for you.” Bless.

When he tells me that his Renfield co-star Nicholas Hoult sent him a picture recently of a Cage-as-Dracula Halloween costume, I mention a group of friends in Brooklyn who dressed as different Cage characters a few Halloweens ago: Ghost Rider, the medic from Bringing Out the Dead, the goofy stick-up merchant from Raising Arizona and so on.

“Oh, that’s hilarious,” he says. “I mean, I don’t know if I’d wanna be in that room where they’re all congregating but I like the idea of it for sure. If I was there, I might start hyperventilating.”

Only afterwards do I wonder how many times he has heard stories like that. We bring up our tales of Nic Cage Halloween costumes, or the time he appeared in our dreams, or we recite our favourite monologues from Moonstruck or Wild at Heart in front of him, and still he makes us feel as if this is the first time he’s hearing it.

Would we do the same to Meryl Streep, Denzel Washington or Robert De Niro? I think not. But there is a communal, celebratory quality to Cage’s performances, an absence of caution or cynicism in his jack-in-the-box vitality, that makes us feel he is ours to share and savour.

“I’ve been trying to push myself to get to a purer place of expression,” he says. “If I look back I can see other places where it has happened, like Leaving Las Vegas, but it’s starting to kick into a high gear for me with Pig and Dream Scenario because they’re personal.

I’m putting these feelings from my life into these characters. And with any luck, it’s connecting or communicating with the audience.” He may have called me on this occasion but you’ll never catch him phoning it in.