Are there any English words containing the same letter three times in a row?
The answer is no, because the usual rules of English spelling outlaw triple letters. We put hyphens in words that contain three of the same letters in a row, so as to break the letters up, e.g. bee-eater, bell-like, cross-section, cross-subsidize, joss-stick, and shell-less. A person who flees is a fleer, not a fleeer, and someone who sees is a seer, not a seeer. Chaffinches used to be called chaff finches, but when the two words were merged, one of the letter 'f's was dropped.
Written representations of noises often contain triple letters, such as brrr, shhh, and zzz, but they don't really count as proper words.