Hi,
don't know what type of paper nintendo use to print out game boxes. As far as know, they mostly use paperboard in the past for packaging designs. I think photo satin paper is one of the best types for handmade nintendo box to my knowledge.
As for your printer, here is some advice. Photo paper is unsuitable for use in a laser printer. Use it only for inkjet printers. The ink cost for the laser too high and the prints are bad. Still, the quality from a laser printer, no matter how expensive, will not equal the quality of a good inkjet photo printer. I recommend you the Canon Pixma for photos because the cost of ink is lower than many other brands, you can buy good cartridges for that printer from third parties for even less, the printer has two software selectable paper sources, both photo and document quality is excellent and the printer is available for a good price.
Hope this will help