In the Lab Rat tie-in comic, it's revealed that Chell was chosen as a test subject by Doug Rattmann, despite her file listing her as being rejected. Why? Because, according to her file, Chell is abnormally tenacious. She never gives up. Ever.
In the last moment of the ending to Portal 2, GLaDOS sends up an old friend to join Chell as she leaves for the surface: the Weighted Companion Cube!
And it's the same Companion Cube, charred from being in the incinerator.
The credits song in 'Portal 2' has GLaDOS sing that She no longer wants to kill Chell and lets her leave, even though she calls her fat... again.
And before that, GLaDOS pulling Chell back from the moon portal.
Especially since she could have finally accomplished her original goal of killing Chell (by letting her fly off into space), and yet she chose to save her life and set her free instead.
GLaDOS's genuine relief and happiness to see Chell is okay after pulling her back from the moon.
Debatably, GLaDOS defending Chell against Wheatley's insults - the exact same insults she herself used earlier. "Look at her. She's not fat."
Even better: "And what, exactly, is wrong with being adopted?" Which she immediately spoils by whispering "For the record, you ARE adopted and that's terrible. Just work with me."
Or maybe if you buy both 'GLaDOS is Caroline' and 'Chell was Cave's and Caroline's adopted child' theories, she is saying that it's terrible Caroline HAD to adopt Chell, because that means something happened to her real parents.
Or if you go with Caroline being in the potato with GLaDOS (not GLaDOS herself), and the idea that Cave and Caroline are Chell's real parents, GLaDOS could be genuinely saying that it is terrible, because Caroline had to give Chell up since she was going to be uploaded into GLaDOS. Both Heartwarming and a Tear Jerker.
GLaDOS's response to Cave Johnson's pre-recorded last words:
Earning the "No Hard Feelings" achievement. The defective turret's quiet, desperate plea of "I'm different" and its subsequent thankfulness upon being rescued really make the moment.
Which may turn into tragicomedy if you then accidentally push it through an Emancipation Grill.
The test results were right. You are a terrible person.
It's right behind a corner! You can do it before you even realize it!
The alternative is leaving it behind, immobile and alone. And it's unlikely that it would be called for the turret opera because where you found it is the last place GLaDOS would look for functioning turrets. Fizzling it is arguably sparing it a Fate Worse Than Death.
It's a turret. Being immobile and alone is basically its function in life. I remain convinced it's happy right where it is.
The ending to the co-op campaign. Atlas and P-Body succeed in opening the giant vault, and inside are thousands upon thousands of human test subjects in cryogenic stasis. Even though the only purpose to finding them was that so GLaDOS could resume her human testing, and it's fairly obvious that most of them are doomed to die during said testing, the fact that GLaDOS herself is so unbelievably overjoyed turns it into a genuine heartwarmer.
"You did it. All that testing was worth it. Look at all those test subjects! Think of all the testing!" GLaDOS might be a poor little passive-aggressive A.I. with psychotic tendencies, but damn if that line doesn't sound like she's ready to cry from pure happiness.
The little space-obsessed core at the end of 2 gets his fondest wish, and he sounds absolutely overjoyed.
Which could have been gut-wrenching instead if they had left in some Dummied Out lines.
Space Core: Earth. Earth. Wanna go to Earth. Too much space. Don't like space. Wanna go home.
During the ending credits of 2, GLaDOS sings about not needing anyone anymore (though she could easily be in denial), and that she might delete Chell from her memory. The lyrics for one line read "REDACTED," and the sound becomes a bit garbled, but the words are still easily audible: "When I delete you/Maybe I'll stop feeling so bad." GLaDOS, you big softy!
The fact that, after everything that happens in both games, after all the hell that GLaDOS and Chell have put each other through, after stopping Wheatley from letting Aperture Science go up in a nuclear fireball, GLaDOS honestly, truly, no strings attached, lets Chell leave.
Part of the ending to Portal 2 where all the turrets stare at you as you rise to the surface, and they all sing the most gorgeous song. GLaDOS may claim she still hates you, but the other beings in Aperture Science Testing Facility make it very clear that they are grateful for everything you've done for them.
It gets even more heart-warming when you look at the English translation of the Italian lyrics:
Dear beautiful, my beautiful darling! My child, oh Chell! [pun: "oh heavens!"] For I hold her in esteem... For I hold her in esteem. Farewell to my dear!
My dear girl, why do you not walk away? Yes, far away from Science. My dear, my dear girl? Ah, my beautiful! Ah, my dear! Ah, my dear! Ah, my child! Oh dear, my dear...
This is particularly poignant if you believe that Caroline was Chell's mother and lives within GLaDOS (as mentioned in the credits song), and thus Chell is, in a sense, GLaDOS's adoptive daughter. in an overbearing kind of way.
During the final test for the Hard Light Bridges portion of the Co-Op Campaign, Atlas and P-body are faced with a problem with a relatively simple execution: cut the bridge so that they fall into a portal and get flung onto a catwalk. As it turns out, the velocity can cause them to overshoot the catwalk and fall into the abyss below. What do they do? Use each other as a wall so they stop right on the mark. Shortly after, Atlas hugs P-body, cementing their already developing friendship.
GLaDOS showing what seems like genuine relief at Chell's survival at the end of Portal 2? GLaDOS admitting "I thought you were my greatest enemy, but all along you were my best friend"? And finally - after insisting that she's totally just getting rid of you because it's less trouble, okay - giving back the original Companion Cube?
Oh, thank god! You're all right!
Just the fact that GLaDOS waits until Chell wakes up and it can be confirmed that she's all right before she deletes Caroline. We don't know how long it took Chell to wake up, but it certainly was long enough to fix the chamber.
It had to have been several hours since it's night when you defeat Wheatley, and day when you leave the facility.
Wheatley's apology. From space.
The space personality core was so obsessed with space. It was cute, and really sad once he realized that, once he finally got to space, he couldn't go back to Earth. This , and this 
The song "Love as a Construct" from Portal 2 for the companion cube is very sweet and oddly touching, in its beepy electronic way.
Love As A Construct nothing, ''Exile Vilify'' is the most heartwarming thing in the entire Valve body ofwork. Just imagine it playing at the end of the Lab Rat comic, as Rattmann peacefully goes to sleep...
Similarly, listen closely to the song's tune. It's the same one the turret opera uses.
Yes, that song's associated with love in this game, it's so sweet how the Companion Cube hums it just for you!
And if you listen carefully, you can also hear it being played (very softly) as Chell is losing consciousness after GLaDOS pulls her back in.
Listen to the end song, and listen carefully. Notice a slight difference in the two last "that's what I'm counting on" parts and the very end of the song? Notice how it sounds like two voices singing together?. That's because it is. It's not just GLaDOS singing there... it's GLaDOS and Caroline! GLaDOS actually decided to let Chell's human mother (if you go with the "Caroline is Chell's mother" theory) sing along with her, as GLaDOS kind of considers Chell her adoptive daughter. In short, "Want You Gone" is more than just a GLaDOS ending song, it's the two mothers singing a farewell to their daughter. If that isn't heartwarming for Portal, I don't know what is!
In this case, "Want You Gone" means "I want you out of this shithole of a place and somewhere else." Which every animal parent does: push their children out of the nest.
Which also may explain their hatred for a certain type of animal, what animal is well known for pushing their children out of their nests? Birds.
Another from the ending song: "Under the circumstances I've been shockingly nice." That line makes one realize that GLaDOS really was much nicer in the second half than one would really think her capable of being.
Another point about the ending song: In the second verse, GLaDOS sings, "Now little Caroline is in here too". The next two lines, on the other hand, are "One day they woke me up/So I could live forever". Those two lines aren't sung by GLaDOS... it's Caroline whose singing those. Not only did GLaDOS let Chell's human mother sing parts in a duet with her, she gave Caroline her own solo part to tell what happened to her - being woken up within GLaDOS so that she could live forever. Not only that, but the next line is "It's such a shame the same will never happen to you", which might in fact be light sarcasm from GLaDOS. Basically that whole part is telling Chell: "We're stuck together because of that failed Brain Uploading, and we don't want that kind of thing to happen to you, so go and live your life freely". Heartwarming and a Tear Jerker when you really think about it.
This live-action short film. At first, it just seems like it's about her having GLaDOS flashbacks and trying to survive, but then she finds the cake book and sheds tears of joy. She spends the rest of the movie baking cake. Even better is what she writes on her wall.
"I get to have some cake today! The cube is watching me. I hope it's proud to see me now, Outside, Alone but free!!"
The ending of the new DLC "Art Therapy" co-op course, where GLaDOSadopts the baby birds. Even though the scene is tinged with the same abrasiveness and fridge horror Gla DOS usually brings to social interaction, her bending down over the incubator as a doting mother is very sweet.
But what really makes the scene: did anyone else think of Chell while watching the one bird peck at its glass enclosure?
Before that, GLaDOS congratulating Atlas and P-Body for shooing the bird out of the facility. She seems genuinely proud of them.
Nice hustle... kill machines.
Chell running into her old science project. It might very well have been the last thing she actually got to do in her normal life. And now, all those years later, after being put through hell, Chell sees her little potato again - and it's well on its way to becoming its own ecosystem. There is no way that didn't cheer her up at least a little bit.
Cave Johnson's pre-recorded message where advises the homeless test subjects to not waste the sixty dollars they'll be getting later. Granted, the message is superficially just as insensitive as any of his others, but there's literally no reason for him to have recorded this besides being, on some level, concerned for his test subjects' well-being. It's a subtle touch that goes a long way to make his character more sympathetic.