Can one Mac mini run twelve displays?
Not if you’re trying to do something different on each display.

If you’re just looking to mirror the same content on twelve displays, then it should be okay: you just need a bunch of DVI splitters such as this one on Amazon.

To elaborate somewhat: the Mac mini is probably the least powerful Mac you can buy today (putting aside refurb models). It only has an integrated graphics processor, and while the Intel&nbsp;HD Graphics&nbsp;4000 is better than some of the graphics cards Apple has used in the past, it’s still no substitute for a proper graphics card.

If we look at the Mac Pro tech specs page:


  Support for up to six displays
  
  Connecting more than three displays requires installation of two ATI Radeon HD 5770 graphics cards.


Even though it hasn’t been updated properly in several years, the Mac Pro still outstrips the Mac mini considerably. You might be able to fiddle it somewhat and get an extra display or too on top of what Apple cite, but I think twelve would be a struggle on a Mac Pro. Doing it on a mini is probably impossible.

Another comparison: the Verge tested a Retina MacBook Pro, and were able to drive three external displays with it (four total). That’s one of Apple’s most modern Macs, and it was struggling:


  There’s certainly a performance hit overall, and when trying to play four separate videos at once on all four displays (one of which was the 4K footage), we did notice the frame rates begin to drop.


Note that three external displays is more than Apple specifies for that machine. (Two running at 2560 by 1600 pixels.)

Driving twelve individual displays on a Mac mini is almost certainly impossible. I would guess that you probably can’t get a Mac from Apple today that will do it, even with a lot of work.

You can add graphics processing power, but eventually it won’t be your graphics that are the bottleneck. Trying to draw windows across twelve 1080p displays (nearly 25 million pixels) will strain what the window drawing tools in OS&nbsp;X are capable of. I suspect memory and CPU power would rein you in.

(That’s not to say that you can’t get computers that will drive twelve displays: you can buy such machines for trading floors, or maybe build a Hackintosh that will do it, but I’m fairly sure Apple don’t cater to that market.)

Also, what on earth are you trying to drive twelve displays from a single Mac mini for? If you really need to drive twelve displays, you need to buy several machines.