Sexaroids Sylvie and Anri, in Bubblegum Crisis. However, given their intended function, this isn't so strange. Perhaps more unusual (and disturbing) is the fact that, of the two Boomer models designed to resemble women, only those purposely designed for sexual use appear to be sentient. And then there's Anri, who's apparently built to look permanently underage...
Minerva-X from Mazinger Z was a Humongous MechaFem Bot was conceived by Dr. Kabuto to fight alongside Mazinger-Z since he whole-heartedly believed a man always must fight with a strong woman by his side (he was a big defender of the Battle Couple trope). Her role was devising a battle strategy to help and support Mazinger according information Mazinger's main computer sent. Okay. However, apparently Dr. Kabuto also thought she needed being capable to think independantly, act on her own and feel emotions to carry out her mission. Not only that, but also he programmed her to love Mazinger-Z (he REALLY was a big defender of the Battle Couple trope). This is reinforced in Shin Mazinger Zero where she feels happiness, uncertainness, worry, jealousy, doubt, despair (to the point of telling her "heart" was breaking on one occasion), joy, and sometimes she even blushes.
Dr. Hell sometimes believed human-resembling robots to infiltrate in the Institute. Obviously he would want they behaved like humans, so it was kind of justified. However Erika forgot she was a robot, and she constantly prayed because she felt empty and soulless.
The comic relief robot in Uchuu Senkan Yamato (also known as Star Blazers) is apparently programmed specially for sexual harassment, though exceedingly nonhuman in form, vaguely resembling R2-D2.
I.Q.-9 (Analyzer) claims that, because of his larger mental capacity, he actually has a wider range of emotions than a human being. "I have more emotions than you." And his little soliloquy after Nova rejects his love is actually very sad.
Alpha from Yokohama Kaidashi Kikou eats, sleeps, dreams, cries, has fantasies and generally behaves in a very human manner, including having a romantic relationship (with another Robot Girl, no less). She never ages though while all the human beings around her do, making for quite some melancholic moments, especially in the manga. The robots of Yokohama Kaidashi Kikou have in particular one feature that is especially ridiculously human: for some reason their data input ports are all in their mouths, which means that for one robot to transfer data to another, they have to kiss. Alpha even has an allergic reaction to milk products.
As it would turn out, this is a rare example of a character trope being invoked. On a careful read through, it can be seen between the lines that the robots were all made to be the final "children" of humanity and to essentially take our place on Earth as the human race was slowly beginning to twilight.
The androids from Armitage III are actually ranked according to how human they are. "Firsts" are basically non-human robots, "Seconds" are androids, and the "Thirds" are so close to human they can get pregnant.
The Cyberdolls in Hand Maid May are hinted to be capable of getting pregnant.
Nearly all the androids in Ghost in the Shell fit this, but generally only in appearance and behavior. The most humanlike robots in terms of emotions, behavior, and just maybe sentience are the Tachikoma's, spider tankettes with bubbly personalities.
As well as GitS, Shirow Masamune is quite fond of realistic robots. At least, enough so to cause confusion in the enemy when they're battle bots... The attack gynoid of Black Magic M-66 may not fool anyone once its taken a bit of battle damage, but it's got a head and hair right out of the uncanny valley, a bosom and feminine curves.
Ghost in the Shell: Stand Alone Complex gives a much greater role to the tachikoma than they have in the manga. Though their physical shape is more close to a Spider Tank and doesn't have any resemblance to humans, their advanced AIs have developed to the state of 5 year old children. In an interesting twist, humans in GitS are almost always shown as emotionally cold, withdrawn, and even mechanical, while the tachikoma are full of curiosity, compassion, and optimism.
This also applies to Real Drive, where android Holon is so human-like that one of the main characters falls in love with her—and dumps his girlfriend who is the one after which Holon has been modeled. Interestingly, Holon mentions that she has no gender identity; she's a sexually neutral sentient AI - but does seem capable of falling in love with a human, or at least forming a strong emotional attachment.
And there's of course the Puma twins from Dominion/Tank Police... Leona is actually shocked to discover they are androids (rather than genetically or surgically engineered humans, presumably) while their artificial nature is a plot point in the sequel manga, Conflict 1: No more noise. Despite being fully aware of their mechanical nature, their behaviour is emphatically emotional and 'human'.
The total inversion of this trope occurs semi-regularly as well. "Jameson-type" cyborgs are nothing more than a small metal lunchbox with four legs and a single, telescoping robot arm on top. Human brain, human legal status, completely inhuman body.
You could replace all of the persocoms in Chobits with spiders with a voice-box, but spiders couldn't perform any of the household chores for which persocoms are shown to be used, or be employed in stores, or provide "companionship" (in any sense of the word) equivalent to that provided by a human.
Good old Astro Boy inspired an entire country's culture with regards to this trope. Despite varying levels of humanoid physical appearance, robots have their own society and culture and even actual robot churches.
Magical Girl Lyrical Nanoha has the Wolkenritter (plural) and Reinforce, programs (albeit magical ones) with emotions and individual personalities. In addition to having physical forms, they eat, sleep, cry, and bleed. The characters of the series pretty much consider them as humans.
Also, there are Agito and Reinforce Zwei, who are functionally anthropomorphic magical wands taught proper Japanese. And, if the theories are correct, Lily Strosek from Force may be one of these, too.
The Cyborgs, despite being mechanical. The only different to the other examples above, they age and even gain or lose weight.
Chachamaru from Mahou Sensei Negima!, while she is programmed for certain things (being Evangeline's servant) possesses human traits, mainly emotions like embarrassment (complete with leaking laser fluid from her eyes), compassion (helping little old ladies up stairs and feeding stray cats), and, most obviously, love. Lampshaded by her creator, who can't figure out how the heck that happened, who (taking "hard science" a little to the extreme) takes personal offense that she's forced to consider philosophy to try and figure out what happened (though she resolves to be more of a mother figure to Chachamaru). Might be justified, since she's partly Magitek.
Played with in Outlaw Star. Gilliam II, the eponymous Cool Starship's computer, is entirely sentient with the sole exception that he is incapable of contemplating his purpose in life. To get over this hurdle, he instead decides to contemplate his inability to contemplate his purpose in life. Trippy.
Rozen Maiden has as its protagonists a cavalcade of animate dolls built specifically to kill each other in a There Can Be Only One tournament. This is all in the good, but one is left to wonder why they were built capable to feel pain, grief, fear and loss. Or, for that matter, affection, attachment and remorse toward their sisters. It's little wonder the tournament didn't get anywhere in hundreds of years.
This could actually be the Whopper Effect [from Wargames]: "The only winning move is not to play." By REFUSING to play the Alice Game BECAUSE they love their sisters, in other words GIVING UP their driving goal to spare the ones they love, may be the only way to come one step closer to becoming Alice. [After all, it boggles the mind for any other reason that an ideal such as the "perfect girl" who is supposed to embody love and compassion is based on the destruction/murder of her sisters. If, however, it is instead a test of purity it finally makes sense. "Only by sacrificing your love for love can you become love."]. If this IS true, then the "There can be only one" may = False as well.
In Steel Angel Kurumi, Kurumi and the other steel angels act exactly like humans except for ridiculous power level. And obedience to the special person who woke the angel from hibernationů by kissing.
Osamu Tezuka's Phoenix includes tales that hit on a group of robots who, despite looking like large metal canisters with limbs, connect better with their masters because they seem a little more human than most. They're connected to a hive mind and, when one is executed, the others walk en masse to kill themselves in lava pits. Furthermore, one who is on the moon at the time considers his more human characteristics at length, because it seems odd that he cannot follow the actions of the others. He eventually proves he's more than a robot by murdering his arrogant boss. And somewhere in the mix, we find out that the robots are more than mere machines, since the first one was made from the joined souls of a human and robot lover (who, yes, appears to have had a soul as well, oddly enough).
The robotic members of the GGG in GaoGaiGar could easily be mistaken for Autobots in both appearance and mannerisms. In one episode, HyoRyu and EnRyu both get into an argument over whether it's right to let 9-year-old Mamoru into combat because of his Zonder-detecting ability, while at the same time, both are drinking... something... out of gigantic cups complete with huge bendy-straws.
This is done again after the battle at Jupiter, when Hyoryu, Enryu, Fuuryu, Rairyu, Volfogg and Mic Sounders all drink that same stuff. If one looks closely on the cups, though, there's a small sign on them saying "Oil". Makes sense that the Strongest Brave Robo Team would need to drink oil to work properly...
Suzu in Hotori - Tada Saiwai o Koinegau clearly has a personality and emotions of his own, and is also (despite the fact that his internal mechanical workings are shown on several occasions and he doesn't really seem to have organs) capable of eating and crying.
Imo-chan from Sora O Kakeru Shoujo eats, sleeps, goes to school, has a job as a maid, and belongs to a club that restores old vehicles. Pretty impressive for a pint-sized, flying robot that appears to be designed as a vehicular auto-pilot system.
Yuki the medical sexaroid from The Galaxy Railways looks and acts so human you'd never notice if she didn't point it out.
In All Purpose Cultural Catgirl Nuku Nuku, a Slightly Mad Scientist roboticist makes an inexplicably human looking (and young, female, stacked, and athletic... ok he's just an old perv) robot for some reason, but can't get the AI part to work and bring it to "life". So, he loads it in the car to tinker with at home over the weekend, along with the kitten that's his young son's birthday present. However as soon as the kid's gone squee and given the kitty its first hug, it runs out into the road and gets pancaked. Faced with a bawling child, a ruined birthday, a physically mangled but vaguely-conscious animal and a brainless bim-bot, he does what any self respecting mad prof family man would do... and transplants the cat's brain into the robot and gives it to the boy as a replacement. Voila, three birds, one stone (please don't ask how the life support mechanisms work). Piles of ridiculously human, ridiculously cat-like, cuckoo syndrome / fish out of water / social naivety hijinks ensue. That and gymnastically fighting off the evil corporation now coming after all of them with guns, wanting their very very expensive 6-foot pile of mechanics, microchips and pneumatics back.
Arale from Dr. Slump is a little robot girl built by Senbei with pretty much the specific purposes of this trope (so much so, she passes off as his little sister to the eyes of the Penguin Village residents), even being that the reason why she wears glasses - her eyesight is horrible without them.
Even though Dragon Ball's Androids 17 and 18 used to be human (and arguably still are mostly human), Android 8 and Android 16 are played straight. Both are shown to be highly compassionate, and developed firm enough friendships with the main characters to help Goku take down Muscle Tower and help push Gohan to Super Saiyan 2 level, respectively.
Haruka's robot caregivers in Kurogane Communication are human enough that she considers them family; she also gets embarrassed when one of them walks in on her changing.
Eve No Jikan is centered around a club where commercially manufactured servant androids can go to interact with each other and humans as humanly as possible. The club specifically has a rule against discrimination against one side by the other. They're actually good enough to fool each other without meaning to! Pretty trippy when you consider that by the Asimovian conventions they're programmed to never disobey a human command unless it means bringing humans to harm.
It is implied that many robots have begun developing emotions, but they are afraid to express them around humans, especially considering the radical position of the Ethics Committee.
Parodied by Mechazawa in Cromartie High School, a ridiculously inhuman robot (essentially a giant tin can with arms and legs) who everyone treats as though he were human. In fact, everyone except the main characters (including Mechazawa himself) seems to believe Mechazawa is human, even when he's doing things like giving himself oil from an oilcan. "We don't say those things about Mechazawa!"
When Mechazawa's kid brother appears, it's implied that everyone knows Mechazawa and family are robots, but they consider it rude to admit such. "We don't bash our friends!"
Yuria is ridiculously human, to the extent that she can get a stomachache from eating food that's gone off, to go along with the whole self-awareness and free will (free, at least, to the extent of her programming as a Sex Bot allows, i.e. constantly wanting to have sex).
Companion Autoraves in Ergo Proxy tend to look like this. The only indication that Pino is one is the fact that her limbs are made of metal, but they're usually covered in clothing anyway.
Although most robots on The Big O are clearly mechanical, R. Dorothy Wayneright and her Evil Twin are virtually indistinguishable from normal people. Aside from a pale skin tone and monotonous vocal inflections they look and sound perfectly human, and R. Dorothy had a copy of the personality of the girl she was modeled after that could be activated by her "grandfather". By the end of the series, it is heavily implied that R. Dorothy has developed genuine emotions and a great fondness/love for her partner, Roger Smith.
Saber Marionette J's entire female cast (minus Lorelei) fit this bill. Being a planet with no women, everybody is a clone and the only women are robots. The Saber Marionettes/Dolls are built with a "Maiden Circuit" that allows them to feel emotions and act more human than the rest of the Marionettes.
In one episode, Tiger effectively commits suicide/allows herself to take "fatal" damage rather than back out of a fight and disappoint Faust. When Lime questions her, she insists that she did it "because I have a heart" She gets better
Kikaider The Animation both plays this straight and averts it. Some robots look exactly like a human, others are cartoony looking robots, and some can transform between one and the other.
Casshern, from Casshern Sins, among many others. In between his episodes of wangst, you might forget Casshern is a robot...until he goes Ax-Crazy...
The robots in Karakuridouji Ultimo have been seen blushing, crying, feeling pain, and eating and drinking. Vice has even been said to have a favorite food.
In SD Gundam Force, we have a justice loving gundam whose heart is linked with a human friend via Power of Friendship, a lady man knight gundam, a hot blooded samurai gundam, a bike that's as annoying as an old man, and Mecha-Mooks that have their own society and TV program.
Nano from Nichijou. Take the winding key off her back, and no one could tell that she isn't human. There's also the puzzle of how her creator, Hakase, managed to build a robot that's more mentally mature than herself.
Cutey Honey: Different revival versions are different of course, but if not for her abilities, it'd often be hard to tell that Honey Hisaragi wasn't completely human from the way she acts. With them, it just looks like her necklace is a Transformation Trinket. Most episodes would change little if she weren't a human character with a necklace of awesome.
Justified, since she was a Replacement Goldfish her father built to replace his deceased daughter, and he wanted she lived like an ordinary girl. He did not even told her she was a robot at the beginning.
Averted in Cutey Honey The Live, where Honey is so psychotically cheerful even when she shouldn't be that she's actually quite believable as a robot - has emotions or is good at simulating them, doesn't understand humans but designed to try and get along with them, able to laugh and cry but at a 4-year-old's level when it comes to knowing which is appropriate when. Unfortunately, this makes her not so good at The Masquerade. Passing for human purely based on You Didn't Ask, she doesn't see what's wrong with using her Healing Factor or transformations in front of civilians and even Panther Claw.
In the New Cutey Honey OAV, it's mentioned that she can bleed and be hurt because her creator wanted her to be as vulnerable as a human in order to have feelings like a human. (She can costume-change to indestructible armor if she needs durability!)