Dr. Hell: "Destroy me? How arrogant. A tiny island country intends to destroy Dr. Hell, the future ruler of the world? Alright! I swear I shall crush Japan in merely ten days! And the whole world shall witness the true power of Dr. Hell!"
Sesshômaru is fond of referring to himself as "this Sesshômaru", a common idiom for arrogance which is often translated into "I, [name]" in English. It was actually explained in a subbing note: Japanese traditionally had a certain "noble" way of speaking, so they'd tend to refer to themselves in third person, or similar.
Rin, meanwhile, does the "cute and childlike" variant.
Hana-chan in Ojamajo Doremi Dokkan, a two-year-old with a Plot-Relevant Age-Up. Subverted slightly in that the first time Hana refers to herself as "Hana-chan" in class (and referring to yourself with a -chan honorific is about as babyish as you can get) she gets cruelly mocked by the class Alpha Bitch.
Cheza in Wolf's Rain always refers to herself as "this one". She seems to be more prone to this in the show than the manga, however.
Kagura Hinata in H2O: Footprints in the Sand. She used normal pronouns as a child, and only started once she was forced to assume her sister's identity. Once she outs herself to the town as Hotaru, not Hinata, she still speaks in the third person, but with the right name.
Pino of Ergo Proxy is a cute little Robot Girl. Though it's grammatically correct to use your own name in Japanese, so she uses "I" in the English dub.
In a variation of this, in the English dub of Tengen Toppa Gurren Lagann Kamina would refer to himself in the third person on occasion when making some sort of declaration (which is based on the original's use of occasional "ore-sama"):
Kamina: Hey, hey, hey, hey, hey! Listen up ya walking faces! Having left his homeland, he never yields, never retreats, and never regrets! He faces forward, and never looks back! He's manly! He's tenacious! Kamina of Team Gurren is here to take you on! Let's go!
Uzura from Princess Tutu combines this with a Verbal Tic to make Uzura seem young and cute-zura! (Which means she says things like "Uzura is Uzura-zura.")
While many characters in JoJo's Bizarre Adventure do this once in a while, Big Bad Dio Brando always refers to himself as "kono Dio", roughly meaning "this Dio". Since that sounds rather silly in English, almost all translations of this expression render it as "I, Dio" instead (e.g. "This is the first time that I, Dio, have felt fear").
Minami Asakura from Touch does this pretty often. However, she makes it flow with her speech and doesn't try to be cutesy, making it hard to notice if you don't pay attention or don't know Japanese.
In the Fruits Basket manga, Yuki's fellow student council member Kimi does this, apparently to make herself more Moe.
Nagisa in the Strawberry Panic! manga and light novels. Thankfully, this "cute" habit didn't make it into the anime. (The official manga translation ignores it, too.)
The 20,001 MISAKAs in A Certain Magical Index not only talk in the third person, but narrate what their emotions are. In a deadpan voice, "'I did not detect any irregularities in your brain,' suggests MISAKA with a bit of anxiety." Last Order, who ups the moe, announces herself as the subject of her narration twice, that is, "... says MISAKA, as MISAKA...," etc. No-one ever seems to react or care. When talking amongst each other, they refer to both the MISAKA number spoken to/about and the MISAKA number speaking. Aloud.
Nena Trinity of Gundam 00 would also do this on occasion.
As would Mileina Vashti, probably to emphasize her youth compared to the rest of the cast.
A few characters from Katekyo Hitman Reborn! tend to do this, including Haru (girliness), Lambo (arrogance), and I-pin (foreigner).
Amae Koromo of Saki, who fits both the cute anime and villainous Western types. Depending on her mood, watching this Insufferable Genius of a little girl speak in the third person can be adorable, intimidating, or both.
Turns out, thanks to them outright stating it in Horitsuba when the uniforms for the school festivals Maid Cafe come up, that White Mokona (Soel) is female and that Black Mokona (Larg) is male. That Mokona is Mokona doesn't change, and Soel will always be everyone's favorite Shiro Manjuu.
Hagall from Ah! My Goddess has referred to herself in the third person a few times.
Aoko Nakamori from Magic Kaito, who is 17 years old and talks like a little girl. It's hinted, though, that this comes from her rather low self-esteem: Aoko views herself as childish, so she acts accordingly.
Yotsuba&!: Yotsuba does this a lot (in Yen Press' translation, but not ADV Manga's translation).
I, Dukemon, spoke in a construct based around this, but only in the original Japanese. It carries over from his base form; Guilmon did use some pronouns like boku from time to time, but most of the time he spoke in a standard third person way. This also applied when he evolves to Growmon. Of course, it's a bit different with his Ultimate form: Guilmon is a rather childlike character, Growmon is rather a Man Child, and Dukemon is a royal knight, so it's going from "Me Guilmon not know pronouns!" to "I, Dukemon, shall smite thee!" This also reinforces that Dukemon has shades of its own personality, supported by having a very different-soundingnote except not actually any different voice to Guilmon. It should be noted that while most subtitles render his pronouns as "I, Dukemon", it carries a slightly different connotation to his actual pronoun in Japanese speech, "kono Dukemon"note this Dukemon, showing his specific reference in the third person is more of a humble servant, befitting his standing as a knight, especially in later media where Dukemon actually served a master, rather than the grandiose and self-important sounding "I, Dukemon".
Culumon follows a similar pattern with some 'culus' thrown in.
Neko: Neko wa Shiro no neko de...Shiro wa Neko no Shiro da yo~~ (Neko is Shiro's cat and...Shiro is Neko's Shiro!
Kind of a plot point in an episode of Hell Girl. Haime and Tsugumi meet a Lonely Rich Kid who talks like this and says that "Mina's Daddy left her". It turns out she is the mysterious Mina, and she's a Creepy Doll...
Zeno from Yona of the Dawn talks like this. It's later revealed to have tragic origins: his wife suggested it to remind himself of his own name, since, living forever and mostly alone with no one to talk to him, he is in danger of forgetting it.