Is asking (and allowing) a trivia question like 'Is there an English word containing four consecutive sets of double letters?' trivialising the site?
Personally, I don't think trivia questions like that are useful for users of this site. I also don't care for single word requests, which I put in the same category. I don't feel like these types of questions raise anything interesting about the English language, rather they encourage infinite variations on the same tedious topic (a word with four As, a word with alphabetical letters, a word that has my name hidden inside it but is not an anagram, the antonym of stairs, etc).
They might find a more suitable home on the Puzzling.SE Beta site where they can be geared toward satisfying the gaming urge more directly. It's not that I don't like them; I just don't like them here. I'm glad you asked the question though. I'm interested to hear what the rest of the community thinks.
Though I don't myself find this sort of question interesting, there is no way we can reasonably call them 'off-topic'; they are about one facet of the English language. Where they might run into problems is 'too broad'. It is unlikely that there is just one word fulfilling the criteria, so there are "too many possible answers". "Narrowing the answer set" could be done by excluding unassimilated foreign words and proper names; but unless you use standard dictionary criteria (whatever your standard dictionary may be), it's purely arbitrary, and the question, as KitFox says, would fit better on Puzzling.SE.
But if there is a wordplay question that does not fall foul of the published close reasons, I certainly don't think we should alter the reasons just to exclude it.