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In "The End", Lister's actually touching response to Rimmer's existential crisis about being a simulation with his memories and thus not really "him"; pointing out that it doesn't matter and that while he might be dead, the fact they're talking proves he's still alive in some form and thus not truly dead in his opinion. Lister then proceeds to treat him exactly like he would have when he's alive, by immediately calling him a smeghead...
Lister would rather forfeit his entire wages and go into stasis for 18 months than turn over his pregnant cat and watch as it got destroyed for breaking the quarantine, ultimately spending 3 million years in stasis due to the drive-leak. The Cat's religion wasn't exactly wrong in their belief that "Cloister" sacrificed himself to be frozen in time, so that their entire race might be allowed to live.
In the episode "Waiting for God", Lister — who has discovered that the race of Cat-people who evolved from his pet cat worshipped him as their God 'Cloister the Stupid', has discovered a dying, blind cat-priest who has lost his faith and succumbed to bitterness. Making a grand entrance as Cloister, Lister ensures that the man goes to his grave happy that his God has finally validated his faith:
Priest: (despairing) You tested me, Cloister, and I failed...
Lister: No, you didn't fail, old man. You passed. I'm giving you an A-plus Distinction.
Priest: You mean... there's a place for me on Fuchal?
Lister: A place? You got your own bathroom ensuite. Corked floors. Your own barbeque on the patio, double-glazing. A phone, everything.
Priest: This is the happiest moment of my li-(dies).
The fact that despite how annoyingly selfish he is, Cat still visited the Priest often. Sure he was pretty clueless after the old man died, but it is Cat.
It's brief, but in "Kryten", Kryten gives a description of a dream he has to Lister about a garden with growing plants. It's sort of cute.
In "Thanks for the Memories", the crew celebrate Rimmer's deathday. Afterwards, a drunken Rimmer tells Lister that he only ever made love once in his entire life. Lister decides to fix things, and adjusts Rimmer's hologrammatic memory...
Rimmer: You gave me eight months of your memory... as a present?
The moment in "Timeslides" when Rimmer and his childhood self salute each other. I mean, knowing that that poor little kid grows up to be the pathetic, weasely man we know today always gets me down. It gets worse when in "Me2", when Rimmer gets an exact duplicate of himself, and they hate absolutely loathe each other. But seeing Rimmer and himself as a kid give each other their own little salute, showing that they respect one another, it gets me right in my heart. Every time.
When Rimmer first discovers a way to change history again. His first thought is that his actions will bring back Kryten and The Cat, he does care.
In the episode where Kryten is to be replaced by the latest android model Hudzen 10, the crew, hung-over as they are, vow to fight to protect Kryten. Even the normally selfish Cat. Kryten muses if this is the human quality called "Friendship". Lister says "Don't give me the Star Trek crap, it's too early in the morning!"
Also Kryten's entire "night to remember," if only because it's one of the rare times when all the Dwarfers are working toward one goal without bickering, something that's usually damn near impossible, in order to give Kryten a decent farewell.
Kryten and Camille throughout her title episode.
Kryten: As huge green blobs go, I think you're really cute.
In "Justice" when Kryten calls up Lister as a witness in Rimmer's trial, one of Kryten's questions was "can you think of anyone who thought fondly of Rimmer?", to which Lister replied "me." The computer could tell if he was lying. So deep down, Lister really does care about Rimmer.
Throughout "Dimension Jump", Ace puts up with all of Rimmer's insults. The only time he ever got genuinely angry was when Rimmer made a jab at Lister.
In Holoship, when Rimmer lists all the reasons he considers himself a failure, but Lister actually tells him there was nothing wrong with his job or the way he lived his life.
Also from Holoship, when Rimmer says goodbye to the rest of the crew. His speech is half-Heartwarming, half-Funny but the best moment comes afterwards. Lister says a quiet "see ya smeghead" and Rimmer actually gives a small smile.
And ultimately in Holoship when Rimmer, the supposed ultimate self-centered Jerkass, finally proves he really is a Jerk with a Heart of Gold when he resigns his new commission on the Enlightenment in order to allow Nirvanah Crane, the officer who had fallen in love with him and sacrificed her place (and therefore her very existence) to let him live his dream life on the ship, to be reactivated and reinstated.
The rest of the crew might be faking their affection for Rimmer at the end of "Terrorform", but they certainly aren't faking the lengths they were willing to go to to rescue him from the Self-Loathing Monster's lair.
Kryten: Sir! Another barrage could bring down the ceiling and bury us all!
Rimmer: You'd risk your lives for me?
Kryten: Of course! You're part of the crew.
When Cat shows loyalty to Kryten over Rimmer in "Quarantine". A far cry from his attitude way back in "Backwards".
When Rimmer (drained of his bitterness and negativity) selflessly throws himself on the grenade in "Emohawk - Polymorph II". He survives the small explosion, but did so without knowing he would. Also counts as a CMoA.
Stoke me a Clipper. There are quite a few little moments in this one, but to my mind the best is at the end, when the crew are having a funeral for "Rimmer" (actually Ace) and Lister awards him a posthumous First Officership. Then all the crew perform a Rimmer-salute.
Adding to this is what what he's awarded the officership for: succeeding in his mission to keep Lister sane. After dozens of episodes of them butting heads and insulting each other, it's clear that Lister does (granted, in an odd, twisted way) appreciate Rimmer.
Even better, when Rimmer leaves, he shakes hands with Kryten and Cat. But he and Lister hug.
It's not just that - It's implied that Kryten and Cat genuinely believe that their Rimmer is the one who died but Lister is fully aware of the truth.
And the preceding space cemetery scene. The rings of a planet are made up of uncountable Ace light bees. Lister points out to an unconfident Rimmer that he's unlikely to break the chain. And the music is wondrous. It's an amazing scene, really.
After Rimmer rejects Ace's invitation for a chat, the others keep encouraging him to go talk with Ace. Until they finally stare at him relentlessly in order to guilt trip him. They're not doing it to bully him, they just want Rimmer and Ace to get along.
In general, the whole idea of the episode is one big heartwarming moment. For six series, Rimmer has been a self-hating loser who believes that he's a waste of space and ultimately will never accomplish anything. This episode is where he finally proves, to himself, Lister and the existence itself, that he's capable of making something of himself and making the multiverse a better place for his existence.
Beyond a Joke, when Kryten realizes that he's been able to make something of himself and do so even though his jilted creator had invented him as a Take That at her ex.
From "Nanarchy", the moment Lister finds the very familiar wristwatch device and buffs it's surface to reveal the very familiar image of the long absent Holly.
After the nanobots have restored Red Dwarf (sort of), we get to see the ship again after a more than two year absence. Best of all we get to see it like it was in the first series. Before the show runners switched over to the CGI model.
Hearing the series original opening theme play as Cat inspects the newly rebuilt Red Dwarf.
The moment from "Back in the Red" when Rimmer enters the bunkroom, Lister (now confined to quarters) is genuinely happy to see him.
Lister: (Lister spots Rimmer walk in) Rimmer!
Rimmer: Word's out they're going to throw the book at you, Listy! Followed by the bookcase, and then the library, brick by brick.
The ending of "Only the Good", Rimmer actually returns from the mirror universe, despite the fact that he has everything he's every wanted in life. He's the captain of Red Dwarf, he's respected among the crew, and the ship isn't falling apart. Despite all this, in the end he still goes back to "his" Red Dwarf.
Back To Earth
Lister meeting the two kid fans on the bus. It's adorable watching them reassure him that, despite all faults, he's heroic and cool to them. They also theorize Kochanski's not dead, turns out despite the fact that they were an illusion, they were right.
Every time Holly's mentioned, it's always in a positive way. Showing that all the crew, including Rimmer (in his twisted way) care for the senile AI.
Whilst it is played for laughs, Lister's video pep talk to himself in Fathers & Suns is essentially his rather bizarre way of acknowledging his shortcomings and attempting to better himself.
Especially since it actually sticks. In subsequent episodes Lister is indeed seen studying for the ship's astro-navigation exams.
The revelation that Rimmer's mother gave him the middle name Judas because she was a member of the Church of Judas, which believed that Judas was actually the twin brother of Jesus Christ, who sacrificed his name and reputation to be crucified in his place, so he could help spread his brother's message of peace and compassion throughout the world, and she wanted her son to have those qualities. Made even more touching as the ending of the episode suggests that this belief was actually right.
In the final episode of Series X (The Beginning), Rimmer finds out that the man he thought was his father was not, and that his real father was in fact 'Dungo' the family gardener, causing Rimmer to be struck dumb. Despite being portrayed in typical Red Dwarf fashion, both Kryten and Lister appear to feel sorry and show concern for Rimmer:
Kryten: This is the worst thing that could happen: Mr. Rimmer has always had a hugo ego and no self-confidence.
Lister: Now his ego's smaller than a pixie's right nut after an icy bath. Well ya know, he thought he came from a military family -high flyers, success ingrained in the very core of their DNA- now he discovers he's no better than me.
And then upon Rimmer's return, he reflects that he no longer hears the voice of the man he thought was his father constantly demeaning him and destroying his self-worth, but that "Dungo.. Dennis.. Dad.." would be so proud of everything he'd accomplished in his life. After Rimmer's plan works and he gives the order to return to Red Dwarf, he proudly declares "The slime's coming home."
Which is a great Call Back to the ending of the very first episode of the series, "The End".