A coffee shop chain in South Korea has posters in the form
By itself, more better is either jocular, dialectal or plain wrong. But this may be two separate ideas (MORE) and (BETTER) (“We serve more coffee than our competitors and the coffee and/or our service is better than theirs”). This is reinforced by other advertising stating “More and Better” and “The more the better”.
On several occasions when I have ordered a cappuccino, I have been surprised to be asked “warm or iced?”. To me, a cappuccino is either always hot (not merely warm; I would never order a warm cappuccino in Australia) or at least by default. Cappuccinos also standardly come with cinnamon sprinkled (again, I would never order cinnamon sprinkle). I am trying to get to “without cinnamon but with chocolate”.
Well, there’s that 1990 Spike Lee movie, “Mo’ Better Blues”: https://www.imdb.com/title/tt0100168
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Within a few hours I saw examples relevant to both recent posts. One was by Mapo-gu council, which read: 새로은 마포, 더 좋은 마포 (new Mapo, better Mapo). The second is the relevant part. It translates literally as ‘more good’ but idiomatically as ‘better’.