Subverted in the sequel, when it turns out she's not an AI, but another siren.
Claptrap and others of his make and model may also count, since they behave in a very human-like fashion.
The Mega Man X series slips ever-closer to this trope's existence with every installment, deliberately. This is in direct contrast to the true, pure robots of the original series. It's implied that even Mega Man himself was not truly capable of pure free will (except in the English version of 7, where he almost kills Wily over his specific objections that doing so would violate the first law of robotics. He doesn't care. In the Japanese version, Mega Man obediently lowers his Arm Cannon when Wily pointed this out). As reploids are designed to be as human-like, or at least as life-like, as possible, it makes sense for further advancements to make reploids more resemble and function like synthetic versions of humans and animals. Interestingly, X and Zero themselves, despite being the models from which all reploids originate and the series' primary protagonists, are still essentially more mechanical than human. By the time of Mega Man ZX, the trope starts to go the other way as well, and the line gets so blurred that Humans and Reploids stop making distinctions between each other, so that when Mega Man Legends rolls around, they've forgotten that there is a distinction.
In Mega Man 9, the hero can't tell the difference between his human creator and a robot built by Dr. Wily, so either the Blue Bomber needs new optical sensors or the robots in the main series have gotten past the Uncanny Valley.
Taken further in Mega Man 10, it turns out robots can actually get sick, and not in the standard "computer virus" sick, as in sick like humans with all the symptoms (Sneezing, coughing, fever). The disease is even called "Roboenza" like the influenza virus. The classic series appears to gone beyond the Uncanny Valley, and the distinction between humans and robots has all but disappeared.
In the Battle Network continuity, the NetNavis of the Internet seem to have free will and consciousness, and by the sixth game they can even enter the real world with the help of a special robot shell.
The 1997 Blade Runner video game, like the film, features genetically engineered creations known as replicants, which are almost indistinguishable from humans. They even have implanted memories to make them believe that they were born and had a childhood. More disturbingly, they are completely self-aware and capable of asking questions regarding their own existence and identity.
Lampshaded in the videogame Oni, where the Diabolical Mastermind Muro is torturing (yes, torturing) the android Shinatama for information in a cutscene.
Muro: "Curious. Why bother programming you to feel pain so intensely? Of course pain is a necessary response to certain stimuli, but they could have dulled the sensation or given you a threshold that would limit the extent and depth of your agony... I'm glad they didn't."
Justified, sort of, by Shinatama's original purpose of monitoring Konoko. ("I've seen everything you've seen, felt everything you've felt...")
In Grandia II, Tio is a robotic killing machine build to fight an ancient war, that inexplicably looks like a teenaged Japanese girl (albeit with pale skin and blue hair). Over the course of the game, she becomes more and more human in personality as well, despite there being no reason for her to be programmed with the capability for emotions.
The robots in Scrap Land. The protagonist himself is said to have built himself up from scratch. Somewhat subverted in that, when a human inadvertently reaches the planet, they freak out.
The androids in the adventure game Zero Zone. One of the puzzles requires seducing one of the robots (yes, there is a sex scene). The ultimate goal of the game is to broadcast a song that sends the titular (and female, natch) Zero Zones in heat, explicitly sending them on a rape rampage, getting pregnant, and thus helping human/robot relationships thanks to the newborns. No, it's not hentai. But it is French, if you're asking.
Mostly in homage to the Terminator example, the titular robots in Snatcher are insanely powerful robotic skeletons covered in artificial skin - their Achilles' Heel, in that it turns cancerous far too quickly when exposed to sunlight. While they are incapable of perceiving things which require actual human perception (such as identifying optical illusions), they appear to feel emotions and act quite a lot like human religious fanatics. They run their own hospitals and biotechnology institutions on the sly, while imitating a real human perfectly to the point of being able to bleed and (it is implied) have sex. They even have a religious obsession with the Kremlin, and herald Dr. Modnar, their inventor, as their god.
Subverted in Lune Zoldark's mecha Valsione. Imagine a humongous mecha formed to have an girlish look, long hair, face that can mimic the pilot's expression (Lune's). And it's controlled with the Direct Motion Link which translates pilot movement into its own movement. Needless to say, this robot is like a giant, walking Lune, enough to make the resident Ascended Fanboy Ryusei go Squee!... for the first time (he'd squee the next time he sees a girlish looking mecha. But this one is definitely the most ridiculously human). Unfortunately, unlike Lamia, Valsione is still a robot, meaning it won't have its own consciousness, thus it looks human, but does not act like one.
A large portion of the plot of Xenosaga revolves around KOS-MOS's strangely human behaviour. This is justified, however, as (warning: major series spoiler) she is actually a vessel for the reincarnated spirit of Mary Magdelene, as was intended by Kevin..
A detail deserving attention is MOMO's distress when finding out that, as a realian, her emotions are programed and her "heart is an optional function".
Xenogears has a robot gynoid made of nanites constructed by an ancient civilization named Emeralda. Her technology was a wonder even then. Upon meeting one of the main characters who looks like her creator, her childlike reaction is to call his name a dozen times. Later she gets an upgrade so she starts acting like a teenager instead of a kid.
Xenoblade Chronicles X: Every single human in New Los Angeles, including the player character, is actually a highly advanced android controlled by a human consciousness, while their actual bodies are trapped in stasis in the core of the Lifehold. They're sufficiently lifelike for the Amnesiac Hero protagonist to not even realize this is the case until they get an arm blown off.
Yumemi from the "kinetic novel" Planetarian may still have quite a few robotic quirks (which becomes ironic if you know the circumstances), but her appearance and behavior are human enough to enthrall the protagonist.
Aigis from Persona 3 is a justified case; if she didn't look human, her mind wouldn't self-identify as human, which is required in order for her to be able to summon a Persona (as Personae are physical manifestations of human personalities and emotions). It's Played for Laughs at one point; when the Kirijo Group buys everyone sushi as a celebration following the destruction of the last Full Moon Shadow, the rest of the party starts calling dibs on the better items, to Junpei's dismay - and then Aigis requests the four most expensive items, at which point he snaps.
"You don't even EAT!"
Persona 4: Arenare-introduces Aigis' "sister" unit, Labrys, previously seen in a Japan-only drama CD, who takes this trope even further. She's much more spirited than her sister, displays a wide range of emotions, and even speaks with an accent. There's a reason for this. All early Anti-Shadow Weapons up to Labrys were created using the personality of an Ill Girl that the Kirijo Group somehow brain-mapped.
You wouldn't think the geth from Mass Effect would be humanlike in any way, right? And that's true, until you realize that mechanical screeching sound they make when you kill them is their equivalent of screaming in pain as they die.
Perhaps justified as the war between Geth and Quarian arose when the Geth started asking difficult self-aware/introspective questions the Quarians didn't want to answer and subsequently tried to kill them all; thus, it wasn't that the Quarians made them ridiculously Quarian robots but that the Geth developed it themselves. The other sentient AIs in the game also demonstrate such traits - one broadcasts static over communication frequencies when you destroy it. Translating the binary, however, reveals the 'static' to actually be the word 'Help' repeated over and over.
The second game implies and the third game confirms that the Luna "VI" was the original basis for EDI's program, having lashed out when the Alliance tried to shut her down after realising she'd become sentient. A rather horrified Shepard apologises for having "killed" her, but she reassures them she understands it was self-defense.
The Geth have even developed a religion on their own. One revolving around mechanical Eldritch Abominations, but a religion nonetheless.
In Mass Effect 2, it's revealed that the geth from the first games are actually a heretical offshoot of the main collective. The true geth have a religion of their own, though. We don't see a lot of it, but the central tenant seems to be "every intelligent creature has the right to make their own choices."
There's also Dr. Eva, the Cerberus plant, from the third game. She successfully tricks the personnel (including several very brilliant scientists) into thinking she's a human scientist... but she's really an AI in a very human-looking robot body, revealed when she walks out of a shuttle crash completely intact, except for her human-looking covering having been burned off. EDI later takes over this body to become a Ridiculously Human Robot herself, although her main core remains in the Normandy.
Legion is a unique Geth on an autonomous platform, specifically designed to interact with organics. It even has movable "facial" features that disturbingly correspond to a person furrowing his brow. Legion does that when asked why it repaired its physical damage with a piece of Shepard's armor. Legion has no answer.
Amusingly, Legion tries frequently to defy this trope, insisting that geth "do not experience fear" or similar things. In Mass Effect 3, the Geth siding with The Reapers in response to the Quarian attack was a clear cut case of panic. Legion justifies it saying that for every geth killed, their own intelligence dims (due to networked intelligence)... Just like how somebody doesn't think logically in a state of panic. There are many smaller, similar moments as well, like the aforementioned conversation about Shepard's armor.
GLaDOS from Portal is a perfectly clinical android that also has a personality and even has multiple cores to define its elements. These include such things as "Anger", which definitely makes little sense. Then again, as this is the same company that came up with such brilliant ideas as the "Heimlich Counter-Maneuver" and the "Take-A-Wish Foundation", it's obvious that they're not terribly practical people; and given that it's suggested GLaDOS was developed as an attempt to develop a fuel line de-icing system that somehow ended up also being "a fully functional disc-operating system" and "arguably alive", they apparently didn't know when to stop adding on additional features. She also has an obsession with cake.
Turrets, for some reason, can also feel pain. Throw one into an Emancipation Grill and listen. They also tell you that they don't hate you, are disappointed when they can't function, spent the time between the two games learning to make music and are sorry to see Chell leave.
Wheatley: I shouldn't laugh; they do feel pain. All simulated of course, but, uh, real enough to them, I suppose.
On a technical note, GLaDOS has some variety of human consciousness inside her somewhere, in the form of Caroline. As to why? Recordings made by Aperture's founder Cave Johnson imply that GLaDOS was supposed to maintain the facility after Johnson's death and pursue portal gun testing with an almost single-minded intensity. Plus, it's implied that Personality Cores were somehow connected to Johnson's desire to upload his mind into a computer system, so...
ATLAS and P-body (Blue and Orange), the robots from the multiplayer, were designed to have genders, masculine and feminine respectively (although Wheatley and GLaDOS have a male and female voice respectively, there's no evidence they particularly see themselves as anything but genderless robots).
If this makes any sense, the automated repair functions of the Aperture Testing Facility. There's an almost organic quality to the movements of the panels; it's most obvious in spots where debris is blocking them, as the system tries to force them into place.
The ur-example for Japanese games is probably Multi from the Visual NovelTo Heart, who single-handedly popularized the "mechanical ears" look now commonly found in anime gynoids.
Some of the mobiles in Gunbound are robotic or mechanical in nature, and all of them are unusually emotive. They look focused/angry when charging for an attack, recoil in surprise when they get damaged, and become visibly tired when at low health.
The Shadow Robots in the final two stages of the True Neutral path of Shadow the Hedgehog are Ridiculously Anthropomorphized Hedgehog Robots, and Eggman implies that Shadow himself is a robot as well, although the other endings of the game seem to counter this assertion, and after beating the Perfect Run Final Boss, Eggman admits that he was lying about that and that Shadow is the real thing.
Kunzite from Tales of Hearts... the first one Tales Series ever got. And that's not the only robot the game has to offer (but he's the only one playable).
He's the least ridiculously human Mechanoid in the game. Croaseraph is an Ax-CrazyOmnicidal Maniac. His brother Crinoseraph turns out the same, just flat and subdued about it. Corundum is a Genki Girl turned up to twelve with an addiction to data. Incarose's repeated failures cause her to break down by the end of the game. Kunzite, meanwhile, is a mere Tin Man who slowly goes from a heartless robot to a fiercely loyal and sentimental but still mostly stoic soldier-knight.
Protos Heis in Tales of Graces. This example doesn't quite understand emotion very well, but it does make jokes, desire friendship, eat (complete with taste), sleep, feel pain and look uncannily human. Asbel calls her Sophie.
Psychic Force's Sonia AKA Chris Ryan, Wendy's sister. Although an android, she looks very much human (and hot) and possesses some sort of motherly personality. Later justified when it's revealed that she's created using the consciousness of Chris.
Curly Brace from Cave Story is an elite combat android, and as revealed partway through the game, so is the protagonist himself. They drown if left underwater too long. This is explained as their shutting down to prevent short-circuiting. No explanation is given for how eating a mushroom restores Curly's memories, or how Quote is able to have implied off-screen sex with one of the Mimigasnote not really, but Most Gamers Are Male was invoked so hard that he might as well have. NPCs mention how, years ago, various groups sent squads of similar robots from the surface to claim the demon crown and kill the Mimigas. Rescuing Curly and restoring her memory reveals that she and Quote were sent to destroy the crown.
Starship Titanic: Nicely avoided, several of the robots look distinctly Art Deco, others look like furniture.
However due to their uploaded brains they act fairly human.
Miharu in Da Capo. Perhaps it's best not to think why someone would design a robot girl that's not only capable of sex, but also apparently possesses a hymen and other... you know what? Let's just stop there.
The player character in Innocent Life: A FuturisticHarvest Moon is a robot built by an enterprising professor to help save an island from apparent volcanic doom. Over the course of the game, he learns how to cook, clean, make friends, and watch television (though you never hear him talk).
The robots produced by Dreamfall's WATICorp tend to have this quality, as revealed during the protagonists' visit to their museum, where it's revealed that previous models of their trademark talking-animal toys had been programmed with features such as the ability to pee themselves and ADHD.
Tekken: Alisa Bosconovitch is a Robot Girl that is also equipped with jetpacks and chainsaws as well as a detachable, explosive head. Get past that, and she's a Ridiculously Human Robot with a childish, sweet-hearted personality.
Robots being built to be ridiculously human, instead of being clearly nonhuman slaves, forms much of the plot of both Wonder Project J games.
Robo from Chrono Trigger is a prime example. When the player first meets him, Robo has no real emotion beyond a willingness to help. As the game progresses, you see Robo showing emotions, loyalty and a willingness to perform Heroic Sacrifices on several occasions.
Fallout 3's The Replicated Man sidequest reveals that the Commonwealth have the technology to build those; they look perfectly human, can feel emotions and even have some flesh and blood. The only difference is in comportment: low-grade ones behave in a rather mechanical fashion while high-end ones are this trope in full effect. You're tasked to find a high-end one that ran away and is hiding in Rivet City. It's Harkness, the security chief. He isn't aware of it because Pinkerton gave him a Magic Plastic Surgery and a memory wipe to help him flee the Commonwealth.
These robots play a major role in Fallout 4, which takes place in the Commonwealth. Called "Synths" (short for "synthetic human"), they range from the crude mechanical skeletons of the Generation 1 series, to the "completely indistinguishable from humans" Gen 3's. The latter group are particularly worrisome for the local populace since The Insitute (the faction that makes the Synths) use them as infiltrators, and have even been known to kidnap people and replace them with copies. To make matters even more complicated, the higher-end synths themselves are not mindless tools (at least the Gen 3's and some of the Gen 2's). Some flee from the Institue's cruelty and try to disguise themselves as ordinary people and just live ordinary lives.
It's not just the Synths, either. Some actual robots in the Commonwealth display surprisingly human-like opinions, beliefs and capabilities. Codsworth and Curie, Mr. Handy variants, both act with incredibly human-like personalities. Curie even asks you to upload her into the body of a Synth so she can become more human, and can be romanced after doing so.. The town of Goodneighbor includes KL-E-O, an Assaultron chassis war-droid that has come to think of herself as a woman, angrily correcting anyone who doesn't refer to her as such, and quite convincingly conveying a "sassy" tone through frequent and precise uses of innuendos. Another female-identifying robot, Ada, becomes a companion in the Automatron DLC, asking for the player's help in seeking revenge on whoever unleashed the killer robots that killed her creator.
In Asura's Wrath, all the demi-gods are actually cybernetic in nature (including the even more human-like civilians). You wouldn't be able to tell, because aside from some facial markings, they look EXACTLY like humans.
The CASTs in the Phantasy Star series often approach this territory in both behavior and appearance. In the games that allow for character customization, they can frequently be made to look human enough that their voices are the only dead giveaway.
Roberta Rossum in The Sims 2 for PSP. Particularly when compared to other robots in the franchise, who tend more towards '50s sci-fi designs.
The robot guards in the fourth world of Sly Cooper: Thieves In Time are very human-like (in the personality sense) despite being skeletal wolf robots draped in ragged furs. While eavesdropping on them, you learn that they have kids which they take to the beach, they feel pain, they have names, have company picnics and play carnival games, and even have a sense of humor. In fact, this is in stark contrast to the otherMooks in the game who seem far more robotic despite being living creatures.
Luna in Virtue's Last Reward- that's whyher name is spoilered. There's room for argument that she's Just a Machine, but not much. She shows compassion for other life frequently, even in circumstances where it would be dangerous to do so. She enjoys being in nature, and (as shown in her private password) loves Sigma. Even her obedience towards Akane isn't ironclad- she can choose to disobey orders, but doing so will mean her deactivation.
Binary Domain uses this trope both tragically and disturbingly. A plot point is that some people are actually robots so lifelike that they don't even know they are one. The stress from finding out causes them to die. In fact they are so lifelike they can have children which are Half-Human Hybrids. Which is all a plan to cause The Singularity.
Accessories in Freedom Wars, which (thanks to being partially made of recycled humans) would be indistinguishable from humans if it weren't for their distinctive outfit, Machine Monotone, and never being more than 20 meters away from the Sinner they're assigned to - and said Sinner gets an extra 20 years slapped on their sentence if they try. The side materials actually explain why they're so human, and even why Sinners get to customize their personal corrections officer: with anything beyond casual friendship with other Sinners being completely forbidden, Accessories also function as an outlet for a Sinner's emotions. They're the one person a Sinner can treat however they like without fear of reprisal.
Star Wars: The Old Republic goes all over the map with this. From the annoying sycophant ship droids, to the Jedi Knight's 300+ year old astromech who has Been There, Shaped History to Dr. Cedrax's "lovely assistant" (and girlfriend) Holiday, to an eccentric Jedi Knight who believes droids are just as connected to the Force as organic beings, to the entire Directive 7 Flashpoint, involving an artificial intelligence that has decided droid liberation should mean killing all humans (and a healer droid who may not like organics, but doesn't want them wiped out, either as Mission Control). But since these are the same writers who gave us Reapers and Geth, it's not surprising.
Shin Megami Tensei IV has Burroughs, the AI provided with every Gauntlet. Despite claiming to lack emotions, she's clearly capable of emoting with her voice—for example, she's clearly excited to tell the other Samurai's Burroughses when you place first in an exploration test, and during the bad ending, there's hints of sadness in her voice as she warns you of the dangers of destroying the Yamato Perpetual Reactor and congratulates you one final time.
The newer Sonic the Hedgehog games have given Dr. Eggman two robot sidekicks named Orbot and Cubot, who, unlike most of his creations, are fully sapient and capable of emotion.
The Talos Principle: All the AI bots, at least those evolved enough to leave QR messages, have very human traits. Justified as making a robot with a human-like mentality is the point of the simulation.
Starbound has the Glitch, who border on exaggeration - despite being entirely mechanical, they need to breathe, sleep, eat and drink, can become poisoned, have two different genders, reproduce sort-of-sexually, can become sick, can become wounded, and patch up their "wounds" with bandages. This is actually justified in-game, though: The Glitch were created as part of a social experiment, and since their long-extinct creators wanted to observe how societies develop over time, the Glitch were designed so that their experiences in life would resemble that of organic creatures as closely as possible. Only a handful of Glitch, including the player character (if you choose Glitch as race), are actually self-aware and even realize they are robots.