For example, I'm writing this sentence: 「かなり無理なお願いだという可能性が高いことを承知しています。」
I can't decide whether the first thing should be である可能性 or だという可能性 and whether the second thing should be 高いこと or 高いということ.
Individually I think I'd feel more comfortable with both being という, but when combined I get a really nasty feeling from there being multiple levels of quotation in the sentence.