The Biltmore library, with approximately 10,000 volumes.
Source.
It's hard to find decent images of the interior of Biltmore House because they don't allow photography inside the structure.
In addition to the books -- which you can still peruse if you are a researcher with permission from the Estate -- perhaps the most noteworthy feature of the library is the ceiling. The painting is called
The Chariot of Aurora and was taken from a palace in Venice.
Here's a
link to a good shot that someone snuck inside the library, showing the ceiling.