Finally learning exactly why River made all those rabbits, especially the blue and yellow one. It was to remind him of his promise to meet her on the moon... which was surrounded by stars that Johnny and River noticed looked like a rabbit.
River's reaction when Johnny tells her that they may be able to buy the lighthouse in a not too far off future. This is one of the very few moments through the entire game when her emotions openly show and swell up so much that she just plainly throws herself into John's arms in overwhelming joy.
The dancing scene in the lighthouse. The bathos that happens when they trip over each other's feet make it more endearing.
In the memory when River and Johnny are riding horses. Even though John is initially hesitant when River is allowed to gallop on her own for the first time it quickly melts away when, and the player can see this if s/he looks closely enough, River laughs. Equine therapy for the win.
The way Neil fiercely tries to prevent Eva from moving River from Johnny's memories in the belief that it would make him miserable is very adorable and proves that Neil does, in fact, care.
Eva: I know. But we are legally bound by the contract to specifically—
When Johnny and River are reunited at NASA. Although this wasn't real, it shows that in his mind, no matter where she was, no matter when or how he meets her, he will always fall in love with her, and that despite their communication problems, she was irreplaceable, and he literally can't picture his life without her.
As heartbreaking as seeing the destruction of Johnny's memories of River was, it's equally heartwarming seeing them reconstructed as they go up in the rocket together.
Meta-example. The Steam version of the game has a lone achievement. It is called "Wish Granted", has the tooltip "send Johnny to the Moon", and a picture of Young River and Young Johnny as the thumbnail.
And the game ties this in just before the credits roll, as Johnny and River, in their child forms, reunite against a white background and Johnny hands the platypus to her before sitting down on the log next to her, strongly implying that they're Together in Death, regardless of the hardships they'd faced in life. Also, this is one of the rare occasions where River smiles.
When Johnny was close to dying prematurely and Eva and Neil were still in his memories.