For games I need the box, period. For consoles and joysticks/arcade sticks I really prefer to have the box and will pay or trade extra for it, but it's not 100% crucial.
I got boxed AV Famicom, Super Famicom, Mega Drive, Saturn, Gameboy Color, Wonderswan, PS1, DS + various clone consoles and newer systems. Currently looking for boxed Videopac G7000/G7400, Amiga 500/600/1200, C64 and PC Engine.
I recently put up an empty Game & Watch Donkey Kong box for auction in a Facebook retro gaming group (lost the game as a kid but still had the box). It sold for US$ 64.00. But that counts as a game box of course