The title character from Bicentennial Man refers to himself as "This one", as do all robots. It is a sign that he has become self aware when he refers to himself as "I".
Kurt Bozwell in Good Burger does this throughout the movie, leading to many funny retorts from the main characters.
Pavi Largo from Repo! The Genetic Opera does this a good fifty percent of the time. Of course, he is a face-stealing narcissist...
Come to think of it, it sort of runs in the family: both Rotti and Luigi refer to themselves in third person a few times as well. Amber probably would do the same, if she ever managed to use a sentence that didn't start with 'I'.
Amber does manage to use sentences that don't start with 'I'—when she's trying to get a hit from Grave-Robber or get into someone's pants.
Or get a hit from Graverobber by getting into his pants.
Data: Data's okay. But Data's tired of falling and Data's tired of skeletons.
Lampshaded and Played for Laughs in The Tuxedo. Jackie Chan's character, who is impersonating his boss Clark Devlin, inadvertently mentions that "Mr. Devlin liked insects too", and tries to remedy this slip by claiming that "I like to refer to myself in the third person." His partner Agent Del Blaine counters: "Del Blaine thinks you're an idiot."
Syd in Children of Men. Syd doesn't know why they want to get inside the camps. Syd doesn't want to know. Syd doesn't care.
Rico in Little Caesar, the film considered the codifier for the gangster genre, did this quite a bit. In fact, his dying words are "Mother of Mercy — is this the end of Rico?"
Machete Cortez from Machete and Machete Kills does this at times, like "Machete don't text." and "Nobody knows Machete."
En Dum En in Dunder Klumpen, much of it is also just statements of the nonsense he think. En Dum En directly translates to A Stupid One so the sentences end up like, "A Stupid One think he's santa. That we are christmas presents in his bag. Happy the child that get me," and "A Stupid One think your nose is a carrot, should feel a little."