Puppy woke me up at the crack of ass," says Jeremy Renner, apologizing for his midday grogginess. It's a gray, wet Saturday in New York City, and we're looking for coffee. He and his dog are holed up at a downtown hotel as he shoots The Bourne Legacy, the rogue-agent tetralogy's latest (and notably Matt Damon-less) installment. "Judo throws," he answers when I ask about his raw, busted-up fingers, which look like melted wax. He's been training hard to get his 40-year-old body ready for the action sequences. This is his first day off in weeks, and despite the combat fatigue and apparent hangover, he's eager to get going. "What else would I be doing?" he says, grinning. "Sleeping, probably."

Renner grew up with a pygmy goat named Sugar. He's the oldest kid, with four siblings who range in age from 37 years to 4 months. He and his best friend (the actor Kristoffer Winters, whom he also confusingly refers to as "my brother") run a successful side business renovating houses. Sometimes he lives in the houses during construction, often without such bourgie comforts as electricity and indoor plumbing. Disciplines he's studied include but are not limited to: world religion, sociology, criminology, Filipino stick fighting, and Muay Thai martial arts. Previous professions: ski instructor, professional makeup artist. He has taught himself to be unafraid of sharks. He has dined with Colin Powell and has regularly basked in the praise of such luminaries as Sean Penn—but about the only time he's found himself starstruck was when he met Cesar Millan, TV's Dog Whisperer. He is, by turns, cut-the-bullshit intense and just-fucking-with-you funny. He's religiously unsentimental ("I don't give a shit about the past") and unabashedly devoted to his cream-colored miniature French bulldog, Franklin.

These are some of the facts that you may collect in your net while standing in the riptide current that is a casual conversation with Jeremy Renner. There are many others that swim by too quickly to catch (what was that about playing with C-4 explosives?).

I'm not saying the dude is weird. I'm saying he contains multitudes. I'm saying he is interesting. Complex. Resolutely present when he talks. Willing to go wherever the conversation takes him. Fearless and frank. When he gets going on the psychology of pygmy goats ("They're all of 12 pounds, but they believe they're a 2,000-pound bull—they'll head-butt you and stare you down like, 'What's up, motherfucker?!'"), you are roped in and down for going halfsies on a herd of the nubbly runts.

There's something compellingly unusual about a guy who's been a working actor for two decades—working but struggling—and then rather suddenly finds himself a leading man with his pick of major franchises. Here he is jumping off buildings opposite Tom Cruise in Mission: Impossible—Ghost Protocol. Next he's leathered up as the arrow-slinging Hawkeye in the Joss Whedon–helmed superhero supergroup The Avengers, alongside Robert Downey Jr. (Iron Man), Scarlett Johansson (Black Widow), Chris Hemsworth (Thor), and Samuel L. Jackson (Nick Fury). Then he's out for vengeance in the just-wrapped action comedy Hansel and Gretel: Witch Hunters. "That was a blast," Renner says. "It's 15 years later and they're pissed off. They're bounty hunters killing witches for a living!"

As we settle into a dark booth at Pete's Tavern, an agreeably ancient and grotty bar near Union Square, some girls at the bar shout at Renner: "Hey, you, I like your stuff!" Really, that's what they say. Renner gives a friendly wave.

He has been nominated for Academy Awards twice—earlier this year as the murderous mad-dog townie Jem in 2010's The Town and last year for his breakout role as Sergeant First Class William James of the Army Explosive Ordnance Disposal (EOD) unit in The Hurt Locker. And before that . . . well, before that he was on an ABC detective show called The Unusuals. It lasted half a season. And even if you'd admired his portrayal of the tortured, torturing killer in 2002's Dahmer, you wouldn't have predicted this for him. You wouldn't have predicted that at 40 years old his dance card would suddenly fill up. You wouldn't have predicted that he'd be playing the lead in the next incarnation of the Jason Bourne films even though his name isn't Matt Damon and a couple of years ago nobody would have recognized his name at all.

It's not supposed to happen this way.

"It is not normal," he says, chewing on each word, an actor fine-tuning a line. "It's not normal," he says again, trying to find the right cadence. "It is . . . not normal."

With that sorted, he orders eggs Benedict and a third coffee. Last night he'd been on stage with the Black Eyed Peas, speaking to a 60,000-strong Central Park audience on behalf of the Robin Hood Foundation. Afterward, he'd run into former heavyweight champion Lennox Lewis ("A monster of a man! That was pretty cool"), then stayed up into the morning at the hotel bar. The "Hey, you" girls shuffle up to the booth and ask if Renner will pose for a picture. He will. They went to Boston College and are, they drunkenly explain, plotting a new kind of social-networking website that's going to change the world. They ask Renner if he's on Facebook. He is not.

"It's pretty interesting," he says of the whole serendipitous and charmed new non-normalcy of being Jeremy Renner. In the old days a Renner birthday party wouldn't have had the star power (and tabloid attention) that his 40th did. Scarlett Johansson was there at his rambling Hollywood Hills house, as was Leonardo DiCaprio, seen getting close with Blake Lively, and Christina Aguilera, who may or may not have fallen asleep drunk in Renner's bed. (The gossips said she did; he told Jimmy Kimmel she was just "hanging out . . . eating cookies and milk.")

"I feel blessed to be working these past 20 years in this business," Renner says. "But when The Hurt Locker came around, it became another thing. And all it was is opportunity. I've been waiting for a role like that for a long time." One of the reasons to cheer for a guy like Renner is his determination to wait it out, keep his head down, and do the work. Waiting for roles that would challenge and matter to him (while paying the bills by fixing houses and doing the odd TV spot for 7-Eleven ham-and-cheese Bakery Stix). Waiting even when there wasn't any reason to hope it might pay off. "I wasn't getting any sign that said, 'This is gonna turn out amazing for you,'" he says. "I wasn't super-happy being so poor. Eating on $10 a month—probably not very good for you. But I loved what I was doing. Not every actor gives their life to do this job. Some just do it as a job. Well, it's my life."