Everytime Rei turns into a weretiger. Not only is the transformation awesome, but the amount of damage he can cause is second only to Ryu's dragon forms.
When Ryu is captured and thrown into the dungeons of Wyndia, he's offered help by the princess, but she's almost immediately kidnapped before she can let him out. What's a six-year-old boy to do? Why, break down the jail door and save her!
During the escape from Balio and Sunder, Nina goes to turn on the lift to the other mountain. While she's doing so, Sunder shows up and goes after her. Despite the fact that Balio and Sunder have kicked his butt four times now, Ryu jumps at him with his sword. Badass.
Garr gets a couple of moments, too. After he inevitably wins the Contest of Champions, he tells Balio and Sunder that he's taking Ryu and his friends as his prize. Balio and Sunder, who up to this point have been unstoppable machines of destruction, try to object, and Garr stares them down so hard they almost wet themselves before agreeing.
It's hard to pinpoint Ryu's personal philosophy, but every time someone in the game tries to convince them that the Brood are evil world-destroyers, OR tries to convince him to turn on his friends, he steadfastly refuses. There are two awesome moments in the game in regards to that:
After the Time Skip, while escaping the mine with Garr, Ryu is confronted by the angry spirit of a dragon, who tells Ryu that Garr and his kind are responsible for the genocide of his race. The spirit demands that Ryu seek revenge and kill Garr. Ryu stops, looks at Garr, thinks about it for a moment, and shakes his head. The spirit angrily attacks in retaliation, and Ryu and Garr kick the crap out of it together (which is also where they start to become True Companions for real).
In the confrontation with the Final Boss, Myria beseeches Ryu to consider all the destruction he's caused, and accept it as evidence that the power of the Brood is too dangerous a threat to the world. If Ryu accepts this as true, Myria wins and the power of the Brood is forever sealed. Canonically, Ryu refuses to believe that it's true (and given his past heroic actions, he's potentially not wrong) and denies it.
During the escape from the dungeon of Wyndia, Ryu fights Balio and Sunder to rescue Nina. There are three possible ways this fight can go. Most obvious, you get the crap kicked out of you. Less obvious, you throw a heal at Nina, who gets up, smacks one of the horse men, then runs. Least obvious? Level grind on Tar Men to about level 16 (takes about an hour or so, depending on how much leveling you did with Teepo and Rei), then fight both of them to a standstill long enough for Nina to recover on her own, smack one of them, then run, leaving the battle to run for another round or two before they determine that they can't beat you, and need to get out of there. Sunder even points out that this isn't how the fight was supposed to go. You can watch the fight here.
After they've destroyed your old life and then relentlessly dogged you for nearly a fourth of the game, smacking poor Ryu with Curb-Stomp Battle after Curb-Stomp Battle, getting to finally put an end to Balio and Sunder is one of the most satisfying RPG villain takedowns ever.