A standard literary practice when most of the Biblical Old Testament was written. New Testament books that are not letters written in direct address use this as well (at least, most Bible scholars believe that John refers to himself when mentioning "the disciple whom Jesus loved").
For that matter, Jesus often referred to Himself as "The Son of Man" while prophesying.
Some Christian scholars believe it was actually a title he used to establish himself as THE Christ and the son of, not just any man but THE man, God himself. This is of course, subject to a great deal of interpretation.
Any number of professional athletes qualify for this trope — Bo knows that Bo Jackson was an early example, and (as with any narcissistic trope) Terrell Owens leaps to mind.
"This is Rickey, calling on behalf of Rickey." — Allegedly, the beginning of a message left by Major League Baseball Hall-of-Famer Rickey Henderson. David Cross spoofed the hell out of this one.
Everyone anxiously awaited Rickey's Hall of Fame induction speech in 2009 to see how many times he'd refer to himself in the third person. He stuck to first person the entire time.
Karl Malone would rather have people remember Karl Malone for being #2 on the NBA's all-time scoring list than for Karl Malone's constant use of this trope, among other examples of Karl Malone's unique command of the English language. Spoofed again, this time by Jimmy Kimmel on The Man Show.
"Comrade Stalin has been told that" Comrade Stalin often spoke in third person. Comrade Stalin also enjoyed quoting Comrade Stalin, in this form.
Rahman "Rock" Harper, the winner of the third season of the U.S. version of Hell's Kitchen, referred to himself nigh-exclusively as "Rock".
Miss Manners refers to Miss Manners in the third person. Judith Martin, the writer of the column, originally presented herself as Miss Manners's "amanuensis."
Famous football player Pelé uses this a lot.
So did Zlatan Ibrahimović in his early years.
Joe Biden sometimes refers to Joe Biden in the third person when discussing Joe Biden's past political accomplishments.
Occasionally comes up in cases of Alter Ego Acting, where the third person and first person may refer to different people. Conversations can get complicated.
The group game Silent Football requires its players to call the other players by their proper names, as pronouns are considered insulting and dehumanizing. Accordingly, many players choose to speak in third person so as not to dehumanize themselves. (In some versions of the game, third person is required.)
Marines' boot camps require new inductees to refer to themselves as "this recruit", instead of "me" or "I", in order to break the trainee from himself and force him to think as part of a group instead of as an individual.
Julius Caesar's autobiographical Commentaries on the Gallic War and The Civil War are written in this fashion. Despite having an ego roughly the size of the Campus Martius, Caesar chose third person to disguise his heavily biased propaganda efforts as balanced, dispassionate histories.
Referring to self as a third person is rather an etiquette rule in Bahasa Indonesia when said self is older than the person who they share a conversation with. Mostly parents to children.
In fact, this is a rather common practice with many East Asian/Southeast Asian cultures and languages.
In Thailand, it is extremely common to refer to oneself's nickname in the Third Person, and there's not even a set rule. Elder to younger, younger to elder, among friends, among couples, among family members, etc. The only rule seems to be only one or two syllable nicknames are used. Referring to oneself's first name or full name is still odd. Note that this is all in informal situations, and in formal situations you naturally revert to pronouns.
Interpreters. They speak about someone else (the speaker) in the first person, and when said speaker speaks about the interpreter, they have to say that in third person.
Wolfman, a DJ for KMAJ in Topeka, Kansas, not to be confused with Wolfman Jack, who also does this, will usually refer to himself in the third person, except when pointing out that he's the Wolfman.
"If the Wolfman says you suck, you suck. So there, you suck."
Averted and avoided like hell in Spanish, especially in the Mexican dialect, because it's normally associated with people with serious mental problems, especially mentally-retarded people. When someone speaks like this in Spanish-speaking media, it's because that person is REALLY stupid or mentally-challenged and in Real Life it's normally used for bullying purposes, not to mention it's considered in some backgrounds discriminatory language.