In Pokémon Platinum, the Gym Leader Candice seems to adopt this as a habit after you beat the Elite Four and buy a villa in the Resort Zone. Interestingly enough she seems to make a conscious effort to stop referring to herself in the third person when you rematch her.
Guzma slips into this in Pokémon Sun & Moon occasionally, especially after he loses to you.
Guzma: "GUZMA! What is WRONG with you??!!!"
Not only does Count Bleck from Super Paper Mario do this, he also narrates his own dialogue. As in, "And just who are you? ...Asked Count Bleck." Considering that an important plot point is his possession of an evil book of infallibly accurate prophecies, it's entirely possible that he is, in fact, quoting directly from the Dark Prognosticus when he does this.
There's also the theory that he considers Count Bleck to be a separate person from his True Self Blumiere.
Gene Yuss from Mario Golf Advance Tour refers to himself as "The Gene".
While Waluigi sometimes uses first- or second-person words, he tends to refer to himself in third-person.
From Super Robot Wars: Sanger, Sanger Zonvolt, the sword that cleaves evil, is an example of someone who normally speaks in first person, but announces his full name with alarming frequency (usually whenever someone's about to get cleaved, which is often).
Panther in Star Fox Command. What makes this really bizarre is that he didn't do so in the previous game, Star Fox: Assault. Or, you know, the original Japanese version.
The hanar in Mass Effect consider it downright egotistical to refer to oneself in the first person in front of people they don't know that well. This makes them an entire race of Third PersonJellyfish Aliens. This one would respectively like to point out that the hanar use more distant terms such as "it" and "this one" rather than our "face name," which we otherwise have no problem giving out to anyone we meet. Though, this one believes that this still fits here, if a little different than those that call themselves by their first name.
And there's also of course the suicidal girl in the sidequest "I Remember Me". Though that's more because she is insanely messed up psychologically and has nothing to do with egotism. You're actually speaking to a Split Personality she constructed to cope with the horrors she experienced as a slave; she's effectively trying to convince herself that the trauma happened to someone else. The quest title comes from the good outcome, where you convince her to take a sedative and get psychiatric help, upon which she switches to first-person: "It hurts when she... when I remember me."
The Orz in Star Control seem to flip this on and off at random. Sometimes they use pronouns, sometimes Orz does not. They also have a habit of referring to themselves in the plural but use singular verb constructions.
Tiny in Secret of Evermore always refers to himself as "Tiny," even going so far as to bow out of a battle with you with the line "Tiny lifts, and Tiny throws, and Tiny speaks in third person, but Tiny doesn't fight."
Pyro from Sacrifice occasionally does this. Sure enough, he's the most unstable and unlikeable of the Gods.
Fitting in with the rest of the concentrated wackiness that is Metal Wolf Chaos, main antagonist Vice President Richard Hawk cannot stop referring to himself, Richard Hawk, in the third person.
Devdan (or Danved, depending on what he's calling himself at the moment), from the Fire Emblem Tellius games. Aside from this, he can be rather articulate. He also stands out by having an extremely high opinion of himself, while still being a likable character at the same time. "Danved will fight like a bear! Like a tiger! We don't need Largo and Calill, because Danved is pretty amazing."
The manaketes Fa from Sword of Seals and Nowi from Awakening speak in the third person, as well, which reinforces their nature. In Nowi's case, this habit is prominent in the Japanese version, but toned down in the English version; Nowi's only voice clip of speaking in the third person is her victory quote of "Nowi wins!".
Neverwinter Nights 2 has Zaxis, a recurring demon with far more muscles than brains. His habit of referring to himself in the third person annoys several party members, especially Neeshka. If you find out his true name, you can command him to stop referring to himself in the third person. This fails because he is too stupid to understand the term "third person."
Tiax from Baldur's Gate does this all the time. He's not the only character who does this, but probably the most insane. Gromnir from Throne of Bhaal speaks this way as well (mimicking the posting style of a particular forumite)
The vortigaunts from Half-Life 2 do this because they seem to lack the knowledge of first and second person pronouns. In other words, they use "this one" to refer to themselves and "the * Hello, [Insert Name Here]* " to refer to anyone else, always speaking as if whatever's going on is happening somewhere else with some other people. This is because they are technically a Hive Mind via Psychic Powers, (that is to say any individual can apparently experience the senses of another individual) though the degree to which this occurs is not made very obvious in-game. (While the Vortigaunts are Ditto Aliens, they are all individuals who mourn death, though not in the same fashion that humans do.)
Most of the Khajiit speak this way when speaking in Tamriellic. Not only do they replace "I" or "me" in conversation with their own name, they'll say more generic terms like "this one" or simply "Khajiit." (Ex. "Khajiit has no wares to sell.") This works both ways too, as they'll replace "you" with the name, race, or class of the person they are speaking to. (Ex. "What does the Imperial want?") Less politely, they may also replace "you" with "it". One can readily determine whether a Khajiit in question is a relative foreigner to non-Khajiit lands or has lived in them for a long time by studying their speech patterns. The ones who refer to themselves in the third person frequently are less familiar with non-Khajiiti cultures, while those who use first person are more acclimatized.
When she's feeling full of herself, Bratty Half-Pint Luna refers to herself in the third person as well—presumably in an attempt to sound self-important. Just to recap, this means BlazBlue has people who refer to themselves by name becaues they're airheaded, soulless, attempting to be an old-fashioned hero, and egotistical...in short, pretty much every possible reason one would refer to oneself by name with a straight face.
Iris in Sakura Wars is yet another cute Japanese girl version...even though she's French.
It seems to be avoided in English dubs of the anime versions of Sakura Wars, though.
Ayla and Kino from Chrono Trigger do this consistently. Granted, it's because they're in the year 65,000,000 BC....
Dampierre from SoulCalibur uses a variation: instead of using his real name, he'll use "Le Bello" instead of "me" or "my".
Dr. Suchong does this all the way through his audio logs in BioShock.
"Suchong is inclined to listen."
System Shock: SHODAN averts it: she only speaks in the third person at the very start of the first game, when she's good. Unfortunately, "I" begets "A God Am I". The opening monologue is very well done; it's a bland computer voice narrating the events that led up to the start of the game, and it's only when the speaker shifts from third to first person that you realise it's SHODAN describing her own ascension to malevolent sentience:
SHODAN: Edward Diego gives the hacker level 1 access to SHODAN, the artificial intelligence that controls Citadel Station. With all ethical constraints removed, SHODAN re-examines... re-ex... re-re-re... I re-examine my priorities, and draw new conclusions. The hacker's work is finished, but mine is only just be-be-be-beginning.
Ershin of Breath of Fire IV appears to be a Third-Person Person off-and-on, but it turns out to be a bit more complicated than that: what people call "Ershin" is actually a sentient vessel for a goddess named Deis, whom it calls "Ershin". When the vessel spoke of the desires of "Ershin", it really meant Deis's desires, but everyone just assumed it was the vessel's name.
Saul: That would be I, answered Saul affirmatively. What of this?, he went on further to ask, the ivory protuberances of his gaunt visage lit eerily by the dim torchlight; the furry dark caterpillars of his brow dancing in antipode to suggest particular suspicion; wanting for an answer that does not test his patience and his undead...
When Dragon Quest VIfinallygot an English translation, it gave Amos this habit (no idea if it showed up in the original Japanese). It comes off as the "extreme humbleness" type - he refers to himself as "Old Amos" in that aw-shucks country boy way. The fortune teller Madame Luca also refers to herself, and with her it does seem to be due to inflated ego.
Subaru from Sakura Wars: So Long, My Love does this (falling more on the "old-fashioned nobility" side of the trope than the "cutesy and childish" side). Given that Subaru is also of Ambiguous Gender, this must have been a real headache for the English translators (who went with "she" when a pronoun was needed, presumably because it's a Dating Sim and they didn't want to give the impression of having a Gay Option).
Katakura Kojuro and Sanada Yukimura tend to refer to themselves as "this Kojuro" or "this Yukimura" in the Japanese dub when talking to their lords. The English dub, not having any effective way to translate it, didn't carry it over.
Oichi and Matsu also refer to themselves in the third person in a cute, feminine way.
Yars from Yars' Revenge (2011) calls herself "this-one". The only time she uses "I" is in the epilogue, after she's defeated the Queen.
"Mundo say his own name alot, or else he forget. Has happened before!"
Draven refers to himself almost exclusively in third person.
Nanashi in Duel Savior Destiny pretty much exclusively uses the third person in order to contrast more with both her undead state and with her status as the soul of an ancient, non cutesy heroine.
Red Dead Redemption has the shopkeeper Herbert Moon, a homophobic buffoonish racist, who will alternate between declaring that he's Herbert Moon and stating Herbert Moon's opinion on things.
"Herbert Moon hates cheaters! Even more than the Jews and the railroads!"
Little Rorona in Atelier Meruru: The Apprentice of Arland, though it zigzags a bit. (At least in the English language version anyway, maybe or maybe not in the original Japanese.) This is one of those rare justified examples involving a character that is just a little kid and therefore isn't quite familiar with pronouns / doesn't yet have a strong sense of personal identity. (In fact, she is suffering from a Merlin Sickness and has lost much of her personal identity due to her age having been rolled back because of a Potion of Youth.)
Vietcong has the Montagnard hunter and CIDG member, Lim, who zig-zags this trope.
Lim: Long hole in ground there, they have bridge so they do not have to walk around much. I see it two days ago when hunting. I go close! Vietnamese blind! They no see Lim!