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Rex: What are you talking about? So what if Van Kleiss is back and more powerful than ever? So am I! If he wants a nanite war, let him bring it!
White Knight: [laughs] It's good to have you back, Rex.
Holiday: ...you know, I think he actually means it.
Pretty much all of "Divided By Six", when it wasn't being awesome.
There are two moments that stand out in particular. The first is when One tells Six how proud of him he is for helping to save the world. The second is when Six tells Rex what convinced him to halt his pursuit of being the most dangerous man in the world...helping to raise Rex.
Made even more heartwarming given how Six behaved before his days at Providence, so it's most likely that One was glad that his adoptive son had grown to be a better person.
In "The Swarm", when Rex 'dies' his life flashes before his eyes - a life that consist of eighteen months by his memory span, but he sees the most important people in his life, including his adopted family at Providence.
A lot of "A Family Holiday" is composed of this towards the end. The most memorable is the speech Six gave to Holiday (with his sunglasses off), their almost kiss, and Holiday meeting her sister as a human.
Holiday:I can see what he did, but... there's no time! I can't... I can't help her. It's over.
(Holiday puts his glasses back on and runs off to fix the machine.)
Six:That's my girl.
And then, after giving Six crap for much of the episode about being the sidekick, Rex gives us this.
Rex: That was... I'm... wow. Six... I've never said it... and I'll probably never say it again, but... I'm honored to be your partner. But don't ever do anything like that again.
The Mood Whiplash moment in "Exposed", where Rex tells Diane to respect the family members of an EVO he was unable to cure.
"Six Minus Six" despite being an absolute tear jerker for the implications for Rex and Six's relationship Six forgot six years, he's known Rex less than two, had some great bits, finally culminating in this exchange:
(Six has Rex pinned and is about to stab him through the chest)
Rex: The smartest thing I've ever done is learn to trust your judgement. And I'll trust it. Even if it's from six years ago.
(Six glares at him and swings the blade down. Rex doesn't move, and the sword stops a millimeter above his chest)
Six: ...Not even a flinch. You believe in me that much, kid? *Helps Rex up* I'd really like to meet this Six you think so highly of.
At the end of the same episode Rex decides to stop them from going through with the procedure to restore Six's memories. Even though it would probably work, the slight chance that it could Mind Wipe him completely was more of a risk than Rex was willing to take if it meant losing his friend and father-figure completely. Not to mention Six, despite having forgotten whatever it was that caused him to make his original Heel-Face Turn, decided to try making (another) one on his own because of the trust and respect Rex had for him.
"Lions and Lambs" is chock filled with these revolving around Breach. When Breach is given time-travel abilities by Van Kleiss, Rex offers to bring her in. He tracks her down and most of what follows is the two becoming friends, and Breach even tries to defend him from Van Kleiss. Things are screwed up when White Knight insists on bringing her in personally, making her lose faith. To get it back, Rex restores her powers (the time-travel device was blocking them). Kleiss thinks she'll use it to send him through time, but she zaps Kleiss instead. Unfortunately, the device goes out of control without Kleiss to shut it down, trapping both her and Rex. Before they vanish, she tells him that, while she isn't sure it was real... she was happy to have a friend. They both hug as the time vortex envelops them.
Rex meeting up with Six, Holiday, and Bobo again after they've gone underground.
Six and Rex have a few brief moments in "Crash and Burn" that show while Six's memory is still gone after the Time Skip, he still cares about Rex in his own way.
Kinda surprised no one's added the part where Rex hugged Six when he found out he was out of his coma, not to mention that Six actually said something positive in response to his reaction. It made this troper's heart flutter and dance.
During a sparring session with Rex, Holiday (lightly) threatens Six if the session were to go wreck any of her equipment, to which Six just answers with a small smile. It's more than enough clue that some things didn't change.
"Double Vision" is one long father/son moment between Rex and Six
At the end of "A Brief History of Time" Breach gets pulled out of the time rift, and dazedly asks Rex if he's real. He assures her that he is.
At the end of the Grand Finale "Endgame" Rex and Cesar manage to make up and hug like real brothers.
Although he says it in his usual gruff, serious tone, there's also the culmination of White Knight's multiple moments of increasing fondness for Rex. He tells Six that their agreement no longer applies, effectively absolving him of the need to kill Rex should he turn into a Humongous Mecha and possibly go out of control. Considering how the series introduced him, that's quite a big deal.