I know of one. Here are some examples:
Elizabeth McDowell." There was reluctance in Yao's voice as he spoke
the name, as if it were a truth he did not wish to acknowledge.
Eliot Pattison, Beautiful Ghosts, p.206, hardcover ed. The first sentence in the chapter begins with a drop capital E and no opening quotation mark.
What's this?" Kleinman asked.
Ethan Canin, Carry Me Across the Water, p. 79, hardcover ed. This section starts with a drop capital W and no opening quotation mark (the book has no formal, numbered chapters).
The Yiddish Policemen's Union,' " says the pie man.
Michael Chabon, The Yiddish Policemen's Union, p. 230, hardcover ed. The chapter beings with a drop capital T and nested (!) opening quotation marks omitted.
The case of a drop cap with an opening quotation mark creates a tension in typesetting/formatting between what may be more pleasing to the eye and what may be more pleasing to the brain.
When the first word of a chapter or section opens with a large raised
or dropped initial letter and the first words are a run in quotation,
the opening quotation marks are often omitted.
Chicago Manual of Style, Quotation marks 13.38
(The 16th edition, 2010, p.633)
This appears to have moved in the 17th edition (CMOS 17, © 2017) to
section 13.37, Decorative initials (“drop caps” and raised initials),
but it’s not free.
There are those who believe leaving off the first quote mark is confusing to readers.
Several years ago, after reading the first, long sentence at the start of a chapter, I was surprised to see a closing quotation mark. I hadn't realized a character was speaking because the sentence made perfect sense when read as narration. Looking back at the start of the sentence, I saw a drop cap with no opening quotation mark. Of course when I reread it, the sentence also made sense as spoken dialog. If this was the norm, why had I never noticed it?
Publishers are divided as to how they handle this case. My impression is that a slight majority choose to omit the quotation mark in their house style guide.
I don't know if some publishers, rather than adhering to a single style guide,
change on some other basis,
e.g. by imprint, series, or perhaps the decision of a book designer.
On the flip side, desktop publishers may have to jump through hoops if they want to keep the opening quotation mark with a drop cap:
How to fix a drop cap with an opening quote mark
Drop Caps and Quotation Marks: A Workaround
Coming Unstuck with Drop Caps
We may have an opinion on this esthetics vs. comprehensibility question, although writers never have to worry about this issue. As readers, however, we might encounter this "other" exception to the "quotation marks come in pairs" rule several times in a single novel without ever taking notice.