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Rizu in A Profile. At one point Masayuki is worried for her when she uses the more standard "atashi" to refer to herself.
Katawa Shoujo's Misha plays with this trope a lot. She sometimes uses what sounds like third person speech, but it's because she's voicing Shizune's signs. She also inverts the trope by speaking in first person for a third person. Lastly, she's not above lampshading it either, which is justified when it's to make clear who's actually speaking.
Da Capo's Miharu refers to herself in third person.
Oko San, from Hatoful Boyfriend, only ever refers to himself as "Okosan". Since he only speaks in subtitled cooing, there's a difference in how the heroine translates for him (pretentious and Hot-Blooded, using the "grandiose" third-person) and how the other birds translate for him (birdbrained and childish, using the "immature" third-person).
Ibuki Mioda in Super Dangan Ronpa 2, though this is toned down in the official English localization where she only does this occasionally.
Fuko of CLANNAD, a small, childish character. She continues to speak this way even when she's an adult, though this because she was in a coma for several years. Her sister Kouko points out that people would take her more seriously if she spoke in first person instead, and Fuuko tries it...but immediately gives up.