It's something his teammates have learned about over time. And something other football types have discovered by observation, and getting to know him.
Along with his rocket arm, plus mobility and high football IQ, the Patriots quarterback exudes another special quality that separates him from others.
It's an inner drive that's unrelenting.
While there are athletes who are so naturally gifted, their path to super stardom comes easy, there are others who might have ability, but need to push themselves to achieve greatness.
Maye is the best of both worlds. The rookie is talented, but he also possesses an internal drive that makes him unwavering in his pursuit of excellence.
On a recent Eye on Foxborough, former Patriots quarterback Brian Hoyer said Maye has a "hunger" to be great, and be considered among the best. That's one of the things that impressed him the most about the first-year player.
Several teammates, who work with Maye every day, echoed that sentiment.
Kendrick Bourne said he sees that hunger, too. The veteran receiver sees how passionate Maye is about making himself better. He sees that competitive fire and determination daily.
"It's his mindset. I think we all should think like that," Bourne said. "But he has that. He has that fire. Everybody's not a superstar, so I put him in that category with superstar potential. Everybody can be great. But being a superstar, not everybody has that (drive)."
Bourne said Maye exhibits it in different ways.
"It's the way he trains, the way he works," Bourne said. "I believe it's going to get him to that point. All of those traits are there. He checks every box."
Most superstars check those boxes. They have that inner drive to be the best. It's undeniable.
Tom Brady had that trait. So did Larry Bird and Pedro Martinez, just to name a few local legends. Many of those icons were driven, uber-competitive and had that unyielding desire to stand above the rest.
Even though he never played with the GOAT, Bourne drew a parallel to Brady, who was highly motivated to prove his doubters wrong.
"That's what I see in (Drake). The way he's making reads, and doing things, he has that potential. That's who I can compare him to," Bourne said. "Like Tom, Drake has that desire to get better and better and better. That mindset of being driven to get better, he has those same traits."
Tight end Austin Hooper had another term for it. He called it "intrinsic motivation." It's in Maye's nature and DNA to push himself.
"You have to be willing to work when the lights aren't on," Hooper said. "It's easy to want to go to work when things are all gravy. But when things aren't going the way you want, you still have that professionalism, that intrinsic motivation, whatever you want to call it, to push forward."
Josh Allen, the quarterback Maye will be opposing Sunday in Buffalo, is similarly wired.
Former Patriots great Devin McCourty, on the latest Eye on Foxborough podcast, recalled how Allen had all the tools between arm talent, athleticism, and physical build to be a superstar.
But he still needed to work to put it all together.
"When you look at the Buffalo Bills, and getting Josh Allen, where he started, how much talent he had, and how it all didn't come together at first," McCourty noted. "He had too many turnovers, it was this, was that. But he seemed like the same kind of guy (as Drake). He just had this competitive nature that he wanted to be great. And you look at how he improved, year in and year out."
Allen is currently the front-runner to be the NFL MVP this year while Maye is trying to finish his rookie season strong and push the envelope in that direction.
"At the end of my career, that last year, it was like, 'Damn, this guy is here, he's arrived, he's owning this division.,''' McCourty said of Allen. "And I think Drake Maye has a chance to get on that same level. He's one of those quarterbacks who have that edge. I think New England has a guy at quarterback that has that same makeup, that wants to get better."
Some of that edge comes from Maye's upbringing, growing up in an athletic family, the youngest of four brothers.
McCourty, now an NBC commentator, got to spend some time with the Patriots No. 3 overall pick during a production meeting earlier in the year. It didn't take long to hear where that inner drive comes from.
"It was cool hearing him talk about his brothers, being the youngest in the family. His brothers always kicking his butt, beating him in stuff, and how competitive that made him, trying to live up to everything they did before him," McCourty said. "He was trying to beat them, he was trying to win. So he definitely has that edge to him."
Patriots offensive coordinator Alex Van Pelt said last week that edge and competitive nature is an important trait for any quarterback, and he was glad to see it in Maye early on. Van Pelt said it's a trait he's seen in some of the other elite quarterbacks he's worked with in his career.
It's also part of what he believes helps a quarterback go from being good to great.
"There's a lot of good quarterbacks. I think the greatness comes with the detail," Van Pelt said. "I think the processing, to me, the great quarterbacks are the fastest processors. They can see the field, they can understand the issues, they can see the space in the field and process that quickly.
"Then the other part, the piece that I've always found with the great quarterbacks, is the competitiveness. Everyone I've been around from [Joe] Montana to [Jim] Kelly, they want to kick your butt in darts or rock paper scissors it doesn't matter. Aaron Rogers, same way.
"So, it's that ability to process. Obviously you have to have the skill set and then it's that competitive nature ... Those guys don't ever want to be wrong. They always want to win and that's that drive and that will."