Back when this thing was new — around 2008 — I purchased a router called a Linksys WRT610N, which boasted a maximum of 5GHz and 2.4GHz on its Wireless-N. It also has ethernet ports, which are much faster and reliable. Another thing it had which was new to me at the time, was a USB port for either a networked printer or storage. This is now commonplace on top-tier routers.
It still runs, and I once had one hard drive (and yes, I mean a HD in a box, connected by USB to a USB hub), with two more thumb drives connected to it.
Recently, as I have been accumulating hardware that I can’t bear to discard, namely 1) a 250 GB LaCie laptop drive; 2) a WD encrypted 1-TB drive; and 3) a Patriot 16 GB thumb drive; 4) an old 512 MB chip that went on an SD-sized drive for a cell phone I no longer use; and 5) my old original 500 GB laptop drive.
I found, after much fiddling, the LaCie is not read by Linksys, so I had to set it aside. The WD drive not only isn’t read, but it seems to interfere with the router, so it too had to be put to one side. Everything else worked, so the answer to the original question in the title is 6 physical drives, to my experience, and possibly more since it might just be a question of compatability.
The new USB hub required for this was a powered 10-port hub, a SIIG (a no-name brand as far as I am concerned).
What I needed to do is to add one physical drive at a time, go back and refresh the web page, declare partitions, then go and connect another one. Sometimes connecting a drive would cause the router to give up its partitions, so you had to re-declare them. But eventually they would all appear, and you can add them as shares in Windows or Linux.