This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Fafner in the Azure: Dead Aggressor
Too many to count. Of note are...
Shouko's Heroic Sacrifice. Especially since it immediately segues into the end credits, which close on an image of other Fafner pilots crying, and the singer going (in translation): "Please God, give me just a little more time".
This is made all the worse in the Light Novel, where the reader isn't even SHOWN the sacrifice, and is only treated to the visit to the grave.
Kazuki admitting that all he ever wanted was to apologize to Soushi for taking his eye out. When he is made to choose to either live or disappear he doesn't really pick one; all he basically says is "I want to really talk to him one last time," which saves him. The way he crumples over the console at the thought that his most important person hates him is utterly heartbreaking.
The way all his negative thoughts just come spilling out of him makes it even worse. "You're probably mad at me. You probably hate me. That's why you wanted me to fight and die, right?" The audience knows that that is not what Soushi feels but Kazuki hates himself so much he actually came to believe that this is how it is.
The scene where Kouyou bids his friends one last farewell before turning into a Master Festum to defend the island.
After Soushi is trapped by an incredibly strong Festum and starts to exhibit signs of disorientation, Kazuki begins to shed tears of distress, shaking Shoushi's 'prison cocoon' and crying his name. This soon evolves into all-out rage and desperation when the Festum kidnaps Soushi. Kazuki propels himself high into Earth's atmosphere in pursuit, screaming:
Kazuki: GIVE HIM BACK! GIVE ME BACK SOUSHI!
Worse? Kazuki doesn't know that Soushi is alive. In the next episode we learn from Fumihiko that he wants to go and avenge Soushi, not rescue him. If he did that he'd definitely die, given how advanced his assimilation is. Kazuki wants to go on a suicide mission to destroy the ones that took Soushi away from him. Wow.
Double worse if you listen to the Gone/Arrive drama CD. Kazuki and his father have a major fight over Kazuki asking to go to the North Pole on said suicide mission and Kazuki breaks down in tears and screams like one wouldn't expect from him of all people. He compares losing Soushi to Fumihiko's feelings after losing Akane. Even Fumihiko can't say no when faced with so much despair coming from his son and allows what we would later know as Operation Soukyuu/Azure.
Mamoru's death. It happens quickly compared to Shouko's death and Kouyou's assimilation, but somehow that makes it more powerful, especially his vision of Sakura and Kenji holding hands right before he dies.
Soushi disappearing at the end of the season, and Kazuki's reaction to it. It's the end of a Trauma Conga Line that leads him into being a Broken Bird in Heaven and Earth.
Right of Left
Just saying "pretty much the whole special" should suffice, but special mention must go to the deaths of main characters, Ryo and Yumi.
"Good Boy. You went straight to your master's side, didn't you?"
The graffti: "Everyone will be gone anyway". A few words that says a lot.
The real kicker of the OVA is that, the executives of Tatsumiyajima are actually forced to put up the L Plan in order to save the island......with the hope of everyone coming back alive (they even prepared enough lifeboats for everyone). Instead they unwittingly send the children to their destruction; the result is longlasting.
Just when we thought that with the end of Heaven & Earth Kazuki might catch a break, Exodus reveals that he only has about three years left to live before the assimilation phenomenon takes his life. Not to mention that Soushi is desperate to save him, and actually chose his career path in hopes of finding a cure.
Chizuru and Fumihiko telling the parents of the third generation pilots that their kids will be going into battle. The expressions of absolute anguish and despair are heartbreaking.
There's a room/shrine dedicated to the people who have lost their lives in the war against the Festum. The walls are covered with pictures, and among them we see Ryo, Yumi, Shouko, Michio, and Mamoru.
Although it appears at first as though he's accepted it, Kazuki doesn't want to die. This is something he hasn't told anyone, and one can only imagine how long he's kept this bottled up.
The thing he wrote on his wish talisman before crumpling it up and saying he hadn't decided yet? "I want to live."
Compared to the guy who only wanted to disappear for years, this is a huge step and utterly crushing.
Canon begging Kazuki not to leave and breaking down when he does.
When Yumiko gets crushed by the collapsing building, Miwa attempts to climb up to reach her, even though all that's left is a hand protruding from underneath the rubble. Then you realize that she's literally a 4 year old in an adolescent body and probably doesn't realize that her mom is dead.
Akira attempts to save a bus full of innocent civilians from a Festum attack. Despite the fact that he gets there in time, everyone gets assimilated anyways and he's left holding what's left of their bloody clothing and a few assimilation crystals. This kicks him into a Heroic BSOD, followed by Broken Tears and a Roaring Rampage of Revenge. Watching and listening to him have a breakdown, especially over the comm as Hiroto is subjected to, is awful. The fact that he is voiced by Yuuki Kaji, a voice actor who does not hold anything back when it comes to emotions, doesn't help.
Exodus (Second Season)
The first episode of the Second Season ended in the most brutal way possible. Hiroto was shot in the cockpit by Human Army's new Fafner and all others around felt someone dropped out of the Crossing System. The fact that Hiroto is waving at the reinforcements only made it more tragic.
In episode 2, while the pilots still have hope that Hiroto is still alive, the viewers are "treated" to a conversation between Dustin and Keith about what to do with his remains. Said remains fit inside a very small cooler. If there was any hope left for us it was brutally crushed.
The human army dropping bombs on unarmed civilians that have already surrendered and Maya being forced to shoot down the Human Army bomber certainly counts.
The decision to travel through Festum infested wasteland is met with whole families commiting suicide. Audiences with keen eyes can link back their happy faces in the last few episodes......and then they went through utter despair and killing themselves.
In an attempt to save Olga from her suicide attack on Walker, Sui attempts to teleport her into his cockpit... only to have her slam up against the wall, re-opening her wounds and ultimately killing her. As if his life didn't already suck enough, this traumatizes him so much that he can barely eat.
Walter's Heroic Sacrifice. Just when he finally makes a sort-of peace with Akira for bombing Tatsumiyajima in the past, and even Akira offers him to go to the island for sorting them out, by the end of Episode 21 he sacrifices himself attempting to save Maya and Akira from being captured by Human Army. Akira's scream really stings it.
Akira again in episode 23 when he sobs over a handful of rice, unable to eat without remembering the thousands of people who starved to death during the exodus.
And again for Akira in episode 24. He sacrifices himself to keep the enemy Festum from assimilating the new island's core. Before he disappears he's reunited with Walter, Hiroto, and others who were killed during the exodus, and breaks down into tears while saying that he knew Hiroto was dead the entire time but couldn't bear to admit it to himself.
Episode 25. What might've been a heartwarming moment when Seri welcomes Maya back turns to this when she looks at Soushi's solemn expression. After a few seconds, her face turns to horror when she realizes that it must mean that Hiroto was dead.
The very last episode ends with Soushi becoming one with Festum. This is a pretty way of saying Killed Off for Real. Sure, he's reborn but put yourself in Kazuki's shoes. This makes it a total of four times that he's lost Soushi in some way. First after their friendship hit the rocks after the whole eye incident, then he gets kidnapped with everyone thinking he was killed, then he actually does get assimilated and then he dies. And poor Kazuki now has to - probably voluntarily - raise a child that looks just like the person that meant more to him than anyone else. Let that sink in.