This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / The Familiar of Zero
Louise tends to give a fairly bad first impression at first, due to her haughty demeanor and her treatment of Saito (who should be noted is not all that bright and tends to have it coming a lot of the time). And then you learn what made her like that. Let's list her issues here...
1. She's the daughter of a legendary mage who is considered THE undisputed #1 Badass in the world for her sheer destructive power and her deeds while in the service of Tristain's Military. In comparison, Louise couldn't really cast any sort of spell properly.
2. Her mother was so strict and rigid that Louise likely never felt much affection from her due to not living up to her expectations, at least from Louise's point of view.
3. She lives in a Mageocracy, where having magic and the ruling class are intertwined, she's all but seen as a worthless defect by her peers AND her society at large.
4. Her parents believed that the only thing she could actually do was to get married to cement political alliances and produce heirs. They were well meaning about it, wanting her to be taken care of... but that sort of attitude wouldn't be conductive to developing a healthy self-confidence.
5. Her being so rabidly jealous regarding Saito? Even if he wasn't what she wanted, her getting a familiar, which is a rite of passage for Halkeginian mages, was literally the first thing that went right in her life after many years of failures and disappointments. And then most of her social group that forms after summoning Saito are also fairly attractive girls that are aiming to get Saito for themselves regardless of Louise's feelings (one of them being HER OWN BEST FRIEND), and don't really hesitate to point out Louise's inadequacies to her face or take advantage of moments when she and Saito are having problems to go in for the kill and attempt to seduce the boy. The only ones not actually trying anything are the other guy in the group, and that guy's girlfriend, and the naive but nice, overly busty half-elf girl that would probably back down if the situation is explained to her. With Friends Like These... DEFINES her social group.
Seeing Tabitha's mother for the first time. She's emaciated, insane and paranoid, and hurls insults and silverware at her own daughter. It's made much worse when the cause is revealed to have been a glass of cursed wine intended for Tabitha (then Princess Charlotte de Orleans) by her Evil Uncle Joseph. Upon realizing what was going on, she drank it herself to protect her daughter, and ended up... well, like that.
Second Season Opening: Saito finds himself back in Japan, then cries that a world without Louise is not worth it. Fortunately, it was a dream.
Second season: The episodes dealing with the Queen's Lost Love, the Prince of Albion. She wants so badly for him to be alive, but in the end we find out that it's just a crude copy, animated with a bit of his soul. The mini-arc ends with her giving his body an Anguished Declaration of Love. Given her role in the series so far, this seems to be a prime example of Break the Cutie.
Second season finale: Most scenes following the reveal that Louise is planning to make a Last Stand / You Shall Not Pass so that everyone important to her can escape Albion (including Saito).
The bit where Louise tells Saito to leave without her... especially when you get a look at her face.
And then when Saito's sleeping potion takes effect right as she's about to finish her part of their wedding vows with the critical "I love you" (though we get to see her mouthing it as she passes out).
When Siesta realizes that Saito used the sleeping potion she had given him previously (with the intent that he drug Louise and run away from the war) to instead take Louise's place as the decoy defense, because, as far as he's concerned, he's more expendable than she is. The poor girl suffers a Heroic B.S.O.D. as a result.
A second later, the full meaning of what Julio said hits Louise and she nearly throws herself off the airship they're on in an attempt to get back to Saito.
There's a very brief Hope Spot when the brightly shining Eclair D'Amour falls out of Louise's robes... only for that hope to be squashed a second later as the flower withers and dies. This scene's potential for Tear Jerkingness is sealed when Saito and Louise scream each other's names... as hundreds of arrows and fireballs fall on Saitou's already mortally wounded body.
Finally, there's the scene afterwards back at the Academy, where all of the named characters appear to be in feeling varying intensities of depression, largely dependent upon how close they were to Saito. Louise is the worst off, staying confined in her dark room, having apparently done nothing but cry since her return from the war.
This troper particularly recalls how she gives an extremely heartfelt 'I love you.' for Saito and questions the cruelty on how she can say this declaration so clearly after he's already gone.
The music that starts just as Saito charges down the hill, sword in hand. Can you place it? That's right, it's a slow, sad, beautiful piano version of the OP, "I Say Yes".
"Agnes, sheathe your sword. He has passed away."
Saito getting back a rush of homesickness after it's been revealed that the contract has been altering his thought process. Played for Drama does not even begin to describe this.
Meta example, Noboru Yamaguchi's death. To make this worse there was a Hope Spot that his cancer was gone.