Michi and Yoshirin, the young couple from Crayon Shin-chan, are a clear parody of this trope. Apart from occasional (and pretty dumb) break-ups, all they do on the series is be lovey-dovey with each other. At least on the original and Spanish dubs.
Yukio Ayashiro and his wife Noriko in a two-part episode of Detective Conan. It's a little more justified as they had just gotten married, tho. Their old friend and the local Private Defective Kogoro openly lampshades the trope:
Just to make it clear, they're both utterly bug-duck crazy: Mika was Seji's stalker until he took a baseball bat to her skull for finding a decapitated head when she broke into his apartment. She got plastic surgery to look like Seiji's true love, making Seiji think Mika was said true love, and they got together. Then the truth came out, but the pair are so crazy they stick together anyway. Oh, and Seiji's true love was said previously mentioned decapitated head, which his incestuous big sister got rid of because she wanted one less rival for her brother's affections (but that's another story)... Did we mention they were crazy yet?
Eureka Seven: Count the times Eureka and Renton argue with each other, come on. That's right, 1. They argued once in a 50-episode show. And that scene lasted shorter than 1 minute. They also tend to say each others names plenty of times.
Hughes has this kind of extreme affection for his wife Gracia and his daughter Elicia, but perhaps atypically, it makes him really likable rather than annoying. It helps that he's a pretty badass Knife Nut.
Sig and Izumi Curtis (of all people) are so lovey-dovey that they traumatized Ed and Al when they were kids. They hang on each other, bombard the unwary with generated hearts, invoke sparkly, Shojo-esque backgrounds, had a Meet Cute which somehow managed being very much that while involving a bear carcass, and frequentlykick ass together while simultaneously outright cooing at each other. Go figure.
Played with when Roy Mustang's devoted aide Riza Hawkeye goes on vacation for a few days, and he proceeds to spend every waking moment on the phone acting like this with his girlfriend Elizabeth. He promises to bring her gifts, speaks fawningly and adoringly, and constantly has little sparkles and hearts roaming around his head. He even admits that the only reason he can get away with this behavior is because his "scary assistant" isn't around to baby-sit him. The twist comes in when it's revealed that "Elizabeth" is Riza all along - she's on an undercover assignment and all the flirting (most of it, anyway) is their very intricate code by which she passes information to him.
Chet and Addy in Haré+Guu. Even the romantically-minded Marie refers to them as "Chetttttt" and "Addyyyyyy".
It's mentioned that every once a Month Adi gets into a fight with Chet over the stupidest things. But always gets back together with him after he says some sappy line.
High School DXD gives us Ise and Rias after his confession in volume ten. The two of them frequently get lost in their own world, staring into each others eyes while ignoring everything around them. In a bit of a twist, the rest of the girls have no problem with this, with one commenting that seeing them together has a healing effect.
Ayano and Shimotakatani from High School Girls are constantly like this, to the disgust of Ayano's boyfriend-less friends.
In Infinite Ryvius, Lilith and Emerson, to the point where they're nicknamed the "Love Love Couple". However, they slowly grow apart over the course of the series.
Yuuki and Yume from Kanamemo, and not even Yuuki's jealousy or Yume's ditziness can put any dents in that.
Lovely Complex: Nakao and Ishihara fit this perfectly. It annoys the hell out of the chronically single Riza.
Umi's parents in Magic Knight Rayearth are more than Happily Married. They take any excuse to fawn and coo all over each other. Even Umi herself is thoroughly disgusted by their incessant, diabetes-inducing displays of affection and refers to it as "the perpetual honeymoon".
Kazuya and Akane at the end of Mai-HiME, possibly to balance that out their counterparts put in an impossible position in Mai-Otome.
Nukoduke has Yuuya and his girlfriend Inori. The two are almost always seen in each other's embrace, and their friends often comment about how well they get along. Of course, the fact that they rarely go out together alone (their mutual friend Yamato is always with them) somewhat detracts the effect of this trope somewhat.
Ouran Highschool Host Club has Tamaki wanting one of these. He has more then just a few Imagine Spot moments where he fantasizes about Haruhi fawning over him, dressing up for him and eating his compliments raw. It's Played for Laughs since sooner or later either Haruhi herself or the twins can be seen in the background, observing him talking to his fantasy idea of her and they always react disgusted, sometimes loudly wondering about Tamaki's mental state.
One-shot characters Oscar and Andi, obsessed with the idea of love, and sprinkle their double battle with hearts, i.e. the attacks of their Pokémon take the shape of hearts. And it is also reflected with their Pokémon: Nidoqueen and Nidoking.
In one Sinnoh episode, a scientist created a machine that would cause this effect between Electric-types and Water-types. Pikachu and Piplup happened to be within range, causing much confusion for Ash's group.
Himeno's father Kaoru and her (very wealthy) stepmother Natsue from Prétear, to the embarassment of their daughters.
In a different episode, Ranma and Akane pretend to be this type of couple to get Ukyo out of the Tendo household. Unfortunately for them, they've made a habit of arguing and find being lovey-dovey with each other a daunting task. But hey, it makes for good television!
In School Days, Makoto and Sekai are called a stupid couple by Hikari.
Sword Art Online: In the opinion of their friends, Kirito and Asuna "can't keep their hands off each other" whether they're in real life or ALO. Said friends are also Romantic Runnerups so take it with a grain of salt.
In Super Gals, Ran and Tatsuki call themselves a stupid couple.
Tokyo Mew Mew: Ichigo's parents couldn't be lovey-dovier if they tried.
Uranborg and Arkangel from the BL manga Yasashii Ryuu no Koroshikata after their Relationship Upgrade. Most of it is Uru constantly fawning over Ark, declaring his love and unending devotion to his "pledged one", especially after they get together. When Ark finally reciprocates, he eats Uru's words up like candy when he isn't getting shy over it. Take a look. (Warning: Spoiler.) 
Katsuo and his wife in Yakitate!! Japan. Their cutesy gushing only encouraged Kuroyanagi to eliminate Katsuo in the semi-finals of one Tournament Arc.
In The Irregular at Magic High School, Mikihiko sees Miyuki giving Tatsuya a shoulder massage, and feels as if he's walked in on something dirty. It's that obvious that the two want to fuck, even if they're not actually doing anything sexual. Miyuki and Tatsuya are aware of this trope and deliberately act even more sickening than usual when they're trying to drive away unwanted guests.
In Destruction Flag Otome Katarina's parents were supposed to be distant and borderline estranged, but Katarina accidentally clears up a misunderstanding about their arranged marriage and Keith's parentage. As a result they become so 'affectionate' that Katarina and the servants have to flee the room partially out of embarrassment at their flirting and partially because it seemed like they were probably going to end up having sex in about thirty seconds.