Anyone knows what a printer that prints 2 mm cardboard A2 size might cost?
I'm better with point sizes, but I think 2mm is about 80pt. Which is super thick. Because of the thickness involved, you're looking at something that doesn't have rollers - I think. Chances are you're actually going to need a digital press. I hope I'm wrong though - digital presses are expensive. We're looking at one for prototyping cards and it's about $15,000 USD.
I suspect you're probably really looking for someone to print on much lighter paper and gluing it to that 2mm board, the way boards for games or cardboard chits are printed. Printing directly on something that thick is, as Gameprinter points out, quite rare.
About the only time cardboard that thick is printed onto directly is via flexographic printing, like on a corrugated cardboard box. It's a much more limited form of printing, and is almost always in one color and of fairly low quality (rarely any screens -- shades of ink -- for example) for good reason.
Correct. The max thickness that people can print on depends on the press type, but 80pt is pretty rare. Our max is about 24pt (.024"), which is pretty common. For an 80pt board/box, we print on a #70 or #80 label stock and glue it to the board/box. Die cut pieces work the same way, but those are often printed on two sheets of 24pt and glued together front and back.