Adrian: WHAT DO I OWE YOU PAULIE!!!? WHAT DO I OWE YOU!? I COOK FOR YOU, I CLEAN FOR YOU, I PICK UP YOUR DIRTY CLOTHES!!!! I TAKE CARE OF YOU GOOD PAULIE I OWE YOU NOTHING!!!! AND YOU MAKE ME FEEL LIKE A LOSER!!! I'M!!! NOT!!! A LOSER!!!
And of course the fight at the end of the first film, where Rocky achieves his real goal of going a full fifteen rounds against Apollo, even though Apollo did win since A title holder pretty much successfully defends if they avoid being KOed. Rocky would have had to utterly dominate the fight to win without being KOed and that would have detracted from the awesome.
It was so awesome, in Rocky II, a handful of people, including Mickey, are convinced that Rocky was the real winner of the fight.
The moment Adrian tells Rocky to win is when we're treated with not one, but TWO training montages! One with Going the Distance, and another with the Gonna Fly Now themes playing.
In the latter, practically all the kids in Philadelphia following Rocky on his run to the steps of the art museum.
Rocky winning the championship the first time. He was a bum from the streets who went the distance with the champ but lost. The fight left him with a crippling injury and his resulting refusals to a rematch left him branded a coward. In order to protect his bad eye he had to unlearn everything he knew and start over in an orthodox stance (something Muhammad Ali said is borderline impossible unless you're a young fighter, and even then it takes years). Meanwhile, the champ (who had poorly trained and underestimated Rocky the first time) pushed himself to defend his title and reputation the best he could. Despite these disadvantages and getting his face smashed to a pulp Rocky was able to knock out the champ and become the heavyweight champion of the world.
A brief moment of Back-to-Back Badasses where Thunderlips strangles Rocky, Paulie jumps in the ring with a chair to attack Lips with it, breaking it on his back, and dared to take him head on. It was all the distraction Rocky needed to take Lips down.
Paulie: I don't sweat you.
The charity match in general. It has nothing to do with the plot (besides foreshadowing a few things, like Mickey's heart condition), but it's a great way to get you pumped up for the rest of the movie.
After the initial training with Apollo goes south, Adrian gets an understated one that's actually rather spectacular in its own right; when she manages to get Rocky to let go and be honest with himself with a hefty verbal beatdown. It doubles as a Crowning Moment of Heartwarming as proof that for all the crap that the two of have to go through, she's there fighting right next to him every time.
Rocky going through training with the additional motivation. He's gotten more mobile, faster, and not to mention, seems to move and fight a lot like Apollo. The Man Hug that occurs at the end seals the awesomeness deal.
After being on the receiving end of Clubber Lang's Trash Talk and getting the stuffing beaten out of him throughout most of the film, Rocky finally returns the Trash Talk toward the end of their fight, goading him into attacking while suddenly dodging and blocking everything Lang throws at him, before knocking him out.
"C'MON! YOU AIN'T SO BADDDD!"
Not to mention how accomplished Rocky felt. He won a match without getting hurt badly, without relying on luck (as Mickey and Apollo likes to claim), and with his own wit to carry him through.
On the meta side of things, during the montage, Rocky appeared on the Muppet show. It was already obvious they used Sylvester Stallone's actual appearance on the show for the footage. However the awesome part is that Jim Henson himself re-dubbed the episode so it would sound like Kermit was introducing Rocky and not Sly.
Apollo's entrance to his exhibition fight, complete with James Brown singing the theme song.
The film itself is full of montages, and probably the most hype-inducing training montage yet.
Duke andRocky, respectively: No pain. No pain. No pain. No pain.NO PAIN.NO PAIN!
By Round 12, as Rocky took more abuse, he was able to stand up against the Siberian Expression, earning the praise of the audience who were once hostile towards him.
Ivan Drago respecting Rocky by Round 15, especially when he decided to turn against his handler.
(translated) "I fight to win. For me!!! FOR ME!!!"
The introduction that plays right after the last film just shows how awesome
Rocky becoming a mentor, and even father figure to Tommy.
This is debatable, but Robert standing up to the bullies that picked on him, and then asking them to be friends is certainly awesome in a sense.
Paulie takes a stand against Tommy Gunn, who insults Rocky. Paulie gave him a verbal smack down, and it pissed him off enough to punch Paulie in the face. Paulie takes it like a champ though.
Paulie: Tommy, you're a piece of garbage, you know that?
Tommy: It ain't none of your business.
Paulie: Ain't none of my business? Ain't none of my business?! The man spit blood for ya, put you ahead of his own family, and you bring these rat bums (GW Duke and company) around here. Huh, Tommy? You got any dignity? Rocko made you your shot, you know that? He's the real champ, and you're just a god damn joke! Now get out of here before I break your head in!
Rocky's Berserk Button got pressed, and now he wants to spill Tommy's blood. On top of that, he denied Duke's satisfaction for a sanctioned fight.
Rocky: (to Tommy) Now you knocked Paulie down, why don't you try knocking me down.
GW Duke: Nah ah, in the ring! Tommy only fights in the ring!
Rocky:My ring's outside.
An out of shape, out of practice Rocky still packs a heck of a punch. He stands up to Tommy Gunn, despite taking a serious beating to the head (which could have killed him or permanently damaged him further), he's still able to beat down Tommy Gunn.
In case that wasn't awesome enough, Rocky walks up to GW Duke, whom Duke threatens to sue if Rocky touches him and laughed at him. Rocky responded by uppercutting him onto a car. That definitely shut him up.
Rocky giving a brief etiquette lesson to a rude jackass and his entourage who won't stop calling Marie a "pig," knocking the punk against a brick wall and forcing him to apologize before roughly shoving him on his way.
"Get a job!"
Ensemble Dark Horse Duke's awesome psyching up of Rocky in Balboa: "Let's start building some hurtin' bombs!"
His entire speech, with all sorts of old-fashioned phrases, manages to sound really badass coming from a trainer with decades of experience.
The training montage in Balboa is a real life one for Stallone. At age 60, he filmed the whole thing for real, all in one day (and notes in his commentary that "moving was not an option" the following day).
The fight is another one for Stallone. He took real shots to the head from the real life Light Heavyweight champion. At one point in the fight Rocky goes down hard and struggles to beat the count (there is a very obvious long count). It wasn't a performance, Stallone had gotten his bell rung and was trying his best to get back up. Again, he was 60 and taking real shots from a real world champion.
This exchange at the start of round 10.
Mason: You're one crazy old man.
Rocky: You'll get there.
Before that, Robert gives a prep talk to Rocky before the start of the final round.
Robert: You have nothing else to prove. People laughed at you and me. Thought of us as a joke. Now, nobody's laughing now!
Then in the middle of it, Rocky gets knocked down, gets a few brief flash backs, before standing up, clearly not going down easy. Then they both just slug away.
In the end, as soon as that bell rang, both fighters hug it out.
Rocky: You're all heart.
The closing credits of Rocky Balboa, featuring real footage of people recreating the iconic run up the "Rocky steps," as has happened every day since the film was released.