Quad chainring (4x12)?
You could do this for the front with a Mountain Tamer Quad (which is an adapter which turns a triple into a quad) and an appropriately chosen front derailleur. The chainline issues become more severe than they do with a triple. 

For the rear, you could use a 12 speed internal gear hub (Sachs Elan) or go up to 14 with a Rohloff Speedhub with a chain tensioner. 

The question is why would you want to do this (like all your questions)? With 3x9's or 2x11's there are more than enough gear combinations that its pointless for riders to have more. With more gears, things have to be machined more carefully and become less robust as well. You need a good gear combination, not the perfect combination (though Nuvinci makes a CVT for bicycles). Also, who in their right mind looks at the cassette when shifting? Nobody. On top of this, front derailleurs are crude and sloppy devices just by how they work -- with 4 chain rings, you can get one to work depending on the chainring spacing, but probably not well.

So, while you could run a 4x12, it would be utterly useless for anyone who wants to ride.