I was wondering if anyone knew of a language (real or fictional!) that did not contain any double consecutive letters (like the double t in "letters").
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Many languages don't allow consonants at the ends of syllables, which means that two of the same consonant will never be next to each other by chance.
Of these languages, some have long/geminate vowels/consonants, or the same vowel twice in a row, and write that with double letters (e.g. Japanese written in Roumaji). So those are no good.
Others either write their geminates differently, like Māori (Polynesian, uses macrons), or don't have length contrasts, like Lingála (Bantu). So those will work for you.
EDIT: Māori may actually have double vowels sometimes! See the comments for details. The point about Lingála stands.
First, let's distinguish between double letters that function as digraphs indicating a single sound (like English "oo" in "flood", which corresponds to the single vowel phoneme /ʌ/, or "dd" in "middle", which corresponds to the single consonant phoneme /d/), and double letters that arise from the incidental repetition of the same sound without any intervening sounds.
There are many languages with writing systems that don't make use of double consonant letters as digraphs. And as Draconis pointed out, many languages don't allow consonants at the ends of syllables, which would prevent double consonant letters from arising incidentally.
But it's harder to figure out whether a language can never have consecutive vowel letters. Double vowel letter digraphs are fairly common as symbols for long vowels. And aside from that, many languages allow vowel-initial syllables, which brings up the possibility of double vowel letters arising incidentally from a sequence of the same vowel in adjacent syllables. (Languages that forbid that kind of situation often resolve the illegal vowel sequence by turning it into a long vowel, which as mentioned might be written with a double vowel digraph.)
If you include sequences of letters separated by a space, that would cause further difficulties, because languages often either don't have or don't write sound changes that eliminate vowel-vowel sequences across word-space boundaries.
If you don't include sequences of letters separated by a space, I think Vietnamese might qualify, because of the general practice of writing a space between all of the syllables in a Vietnamese sentence, regardless of word grouping. In the linked Wordreference thread, the original poster asked if mặttrời would be an acceptable spelling of mặt trời, and was told to use the spaced spelling.
As far as I can make out from the Wikipedia article on Vietnamese spelling, a doubled letter shouldn't be possible in a single written syllable of Vietnamese. Vietnamese does have a diphthong written "ưu", which might look like a double letter to an English speaker, but ư and u are considered to be separate letters of the Vietnamese alphabet.