When it comes to pairings on "The Big Bang Theory," it often feels like one of the most contentious is between Sheldon Cooper (Jim Parsons) and Penny (Kaley Cuoco). As the series begins, Sheldon is dismissive of Penny, his beautiful new neighbor down the hall with whom his best friend and roommate Leonard Hofstadter (Johnny Galecki) is completely infatuated. The two serve as a classic sitcom odd couple, with Penny driving Sheldon absolutely insane by stealing everything from his food to his WiFi to his spot on the couch. As for Penny, she finds Sheldon to be insufferable most of the time, thanks to his social awkwardness and genuinely insulting superiority complex. This evolves throughout the show, though, and actually? The friendship between Penny and Sheldon becomes one of the series' most touching relationships.
There's a decent list of moments that we could include to describe how touching Sheldon and Penny's friendship can be under the right circumstances. Sheldon, usually afraid to trust people, comes out of his shell around Penny, whereas Penny realizes that, underneath the bruff and bravado, Sheldon is often scared (see: how many times she sings "Soft Kitty" to him when he's sick or unwell). The very best moment between them, though, is during the series' 200th episode, Season 9's "The Celebration Experimentation."
In the episode, Amy Farrah-Fowler (Mayim Bialik), Sheldon's girlfriend, decides she wants to throw him a fantastic surprise party. Everyone works to put together the party, and stars like Adam West and Wil Wheaton even manage to make an appearance in Sheldon's honor.
Sheldon enters the party and immediately runs into the bathroom; after Amy and Leonard argue over who should go and talk to him, Penny takes matters into her own hands. She finds Sheldon, seemingly on the verge of a panic attack, telling her the entire experience is simply "too much." Sheldon says he's sorry but he doesn't think he can return to the party, and Penny says, "That's fine."
Perching on the end of the bathtub, Penny says to Sheldon, "You know, I hate that your sister and her friends used to torture you," referring to the revelation that Sheldon's twin sister once tricked him into thinking Batman was coming to his birthday party, only to stand him up. "What I hate even more is that if I was there, I would have tortured you too. My point is, there was a time I never would have been friends with someone like you, and now, you are one of my favorite people. So if what you need is to spend your birthday in a bathroom, I'm happy to do it with you." The two return to the party together — and clearly, Penny was the only one who could help Sheldon in that moment.
Sure, Penny and Sheldon fight a lot during "The Big Bang Theory," but the fact of the matter is that they grow to care about each other — and it makes for a truly touching bond. They both love Leonard, Sheldon's best friend and Penny's eventual husband, and Sheldon even comforts Penny when Leonard is away at the beginning of Season 7, trying to move past his relative lack of emotional intelligence and actually support his friend while her partner is far away and not able to contact her very often.
This all builds to the series finale, "The Stockholm Syndrome," where the whole gang travels to Sweden so that Amy and Sheldon, now married, can receive their Nobel Prize in Physics for their research on super-asymmetry. It doesn't start out well; Penny is pregnant during the flight but trying to hide it until she's ready to tell her friends, but she and Leonard end up telling Sheldon because he's so paranoid that she's simply ill. He then immediately tells everyone, and after he does even more insensitive stuff, the gang almost leaves before the ceremony.
They return, and Sheldon, after being admonished by Amy, dedicates his speech to each of his friends in turn. When he gets to Leonard and Penny, he describes them as his "two dearest friends in the world." It seems impossible to imagine that this mismatched pair would ever really care for each other, but when push comes to shove, Penny and Sheldon are real, genuine friends by the time "The Big Bang Theory" ends.