Two in particular are Cartiere Magnani and Richard de Bas. Cartiere Magnani began production in 1404. Moulin Richard de Bas began in the 15th century as well. Cartiere Magnani still offers mould-made paper amongst its wide portfolio, while Richard de Bas makes paper, in mould, by hand, at a rate of only 200 sheets per day.
Both houses enjoy a fantastic history of use. Moulin Richard de Bas was used for the Constitution of the 5th French Republic in 1958. They have also made the papers for the issuance of the degrees of the Nobel Prize. Cartiere Magnani provided paper to Picasso and was the paper of choice for Napleon's wedding invitations.
So, what makes these papers so valuable that they would be used for the most extraordinary events of western culture? Their history, of course, contributes to their desirability. However, I think that what makes them truly luxurious is that they are often made from cotton, and mould-made, meaning that the paper slurry is dried by hand in a sheet mold, rather than in a long conveyor in web (a long roll of paper stretching from slurry to the dried product). Some of the de Bas papers have floral inclusions (floral material mixed into the cotton). Both mills produce sheets that have four deckled edges (a "deckled edge" is the unfinished edge that is a result and proof of a hand-laid paper, either in a mould or loose on a screen).
In the end, however, I believe that their desirability is tied to their unique manufacture. These paper stocks look and feel like nothing else. Their stability is exceptional. They are anything but ordinary.
So, what motivated me to mention this? Recent use of one of my own favorite manufacturers, a bit more current at a mere 400 years old, rather than 600 years, Hahnemühle. After looking at the extraordinary texture and quality, I felt compelled to look at some of the more ancient stocks, and all of this has inspired me not just to pass it along to you, but to undertake making my own hand-laid stock. So look forward, hopefully, to a post in the near future where we'll make paper ourselves!