As a young teen, Indy is forced to give up the Cross of Coronado and the bandit Fedora turns to him and says: "You lost today kid, but that doesn't mean you have to like it." He then gives Indy his hat, which then becomes Indy's hat, forever.
The scene where Henry Jones Senior was overjoyed that his son did not die in the falling tank, and for the first time hugged him with genuine love.
Henry Sr.:(sobbing) I thought I'd lost you, boy!
Indy saving his dying father with the Holy Grail, and allowing him to finally hold the object that he has spent his life seeking.
Henry Sr. gives him a look as if to say, "Is this...?!"
And then he sets it aside to hug his son.
After Henry saves their lives by his scaring seagulls into the Nazi pilots plane, when Henry muses, "I suddenly remembered my Charlemagne: 'Let my armies be the rocks and the trees and the birds in the sky.'" Indy gives his father a look of newfound respect.
Henry going alone to rescue his old friend Marcus being held in a tank and their greeting each other with the old toast from the University Club complete with silly hand gestures:
Henry: "Genius of the restoration..."
Marcus: ..aid our own re-scu-ci-ta-tion!"
Indy, hanging from a cliff, tries to grab the Holy Grail but is slipping from the edge and his father needs both his hands. "Indiana? Indiana... let it go." Not only an awesome twist on Friend or Idol Decision, but the first time his father called him "Indiana".
Even moreso, the Grail is what Jones, Sr., has spent his whole life looking for and researching. The fact that he values his son more than that has to say something, especially since Indy was convinced that the Grail was the only thing his father cared about for the longest time.
More even than that: Indy spent the whole movie condemning his father for his Grail obsession. Now, he is risking his life to see that Henry's life work isn't lost.
This little exchange at the end:
Henry: Elsa never really believed in the Grail. She thought she'd found a prize. Indiana: And what did you find, Dad? Henry: Me? Illumination.
The Grail Knight standing just long enough for Henry to see him wave goodbye.
The classic riding off into the sunset, with Indy reaching out to steady Marcus on his horse.