Thermaltake Core W200. 14 stock, not sure if you can still get extra internal trays/brackets from TT, anymore (so many drives possible if you can), but you can absolutely add 3x5.25" bay -> 4x or 5x3.5" devices to add 12-15 more. Beyond that, stack it on a TT P200 pedestal and toss in various HDD adapter modules for even more drives.
Or a Phanteks Enthoo 719 with a bunch of stackable HDD brackets in the base and up the side.
The Phanteks Enthoo 719 will hold 12x3.5 and 3x2.5 with the basic layout. I could easily add 6 more 3.5 drives and likely 4-6 more with a little modification. It's a very roomy case with lots of ventilation and fan placement locations.