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Tear Jerker: To the Moon
The whole game is one long one, but the ending scene takes the cake. Other than the scene where Everything's Alright plays.
Alternate!River reaches out her hand to Alternate!Johnny, and Johnny turns around and takes her hand. Everything you've been doing for the past 5 hours comes to this point. And you just break down at that moment. And then the montage of Johnny's new memories rolls, of his wedding to Alternate!River, building their house, growing old together, this time with Joey along with them. At this point you are consumed by grief and joy and hope and despair and guilt and so much relief that you just want to find the people you love and hold them tight.
When you realize that River made those paper rabbits to remind Johnny that they are supposed to meet on the moon if they ever get separated. Only Johnny can't remember that memory because of the beta blockers. But River doesn't tell him of their very first meeting. Why? Who knows. Maybe because she didn't want to remind Johnny of his tragic past. Maybe her Asperger's prevented her from fully communicating with him. All we know is that she diligently and quietly folded those rabbits, trusting him to understand what she meant. And then she died. But Johnny finally fulfills his promise before he died. Maybe. In a way. It was artificially implanted, but it was still real to him in those final moments of his life. Maybe. The only comfort you can take away is that he died happy and doing what he promised, only he didn't really do it. When you realize all this it just breaks your heart into pieces.
The reason why is fairly clear - River was never aware of WHY Johnny forgot his promise, along with her. Her condition was the reason she never asked outright, but she was never aware of what had happened to his brother, because by the time they reunited, he'd had that memory suppressed. On top of that, this adds a huge dose of Fridge Brilliance to what she said at the movies - that she believed that they were watching it together without being physically near each other, just as she'd probably done the next year at the carnival. All that grief because of one tragic event...
He DID remember his promise! He still remembered to go to the moon. He just forgot why.
The implication during The Stinger that Neil might be dying. It becomes even MORE depressing when you realize why he acts immature, mean, jerkish, and does stupid ridiculous things. (like riding a horse backwards) He's trying to get the most out of life before he dies...
Or he's just addicted to pills and is going through withdrawals.
Why does River care so much for Anya, a lighthouse of all things, so much that she is willing to die instead of letting it be alone. Because when they first met, River called all the stars lighthouses that could never reach out to anyone and were always alone, but that one day she wanted to befriend one of them. The implication being that she herself no longer wanted to be alone either. And then you can demolish boxes of tissues when you realize exactly why she asked what she did.
The hackey-sack. It was given to River by Johnny when they first met, but Johnny no longer remembers the event. When he reveals to River the reason he approached her during "their first meeting", she asks him to throw away the hackey-sack. It was probably around then that she realized that he had forgotten about their REAL first meeting. It makes everything afterwards so much more bitter in hindsight.
When Johnny is discussing what he likes about River, his only attraction to her is how she is strange and different from everyone else. And all he knows is that he WANTS to be different, but he can't put why he feels so exactly into words. At this point he has lost his memories of his twin brother Joey, but his mother still calls him by that name and he most likely subconsciously wanted to stand out and not be seen as a replacement. Perhaps even worse is how different this "first meeting" of his with River is from their real first meeting.
The Reveal, when you discover Johnny and River's true first meeting, is so beautiful, tragic, bittersweet, adorable and magical, it would probably be the first point you start crying if you had been holding back tears before this.
And there's a major one before the player even sees River, when Johnny is sitting at their half-finished house on the bluffs and having a dramatic conversation with Izzy. When she criticizes his decision to lie to River for her own health, his retort is heartbreaking.