Nintendo has a long history of innovation in game design; of all the games they have ever made, which do you consider their most revolutionary, and why?

1080 Snowboarding, that's three Revolutions right in the name!

This is a hard one, Super Mario Bros. basically created side scrollers, Metroid created a whole genre, Super Mario 64 revolutionized 3D games, Ocarina of Time revolutionized action/adventure games, then we had that whole VR deal, yes, the Virtual Boy gets a mention here, then they revolutionized motion controls too, oh, and games with touch controls...
Oh, and then they revolutionized open world games with Breath of the Wild.

But their greatest contribution to gaming was Wii Music—

Either:
Donkey Kong (arcade) - first platformer ever, introduced two legendary mascots, put Nintendo on the map, or...
Super Mario Bros. - one of the first side-scrollers and substantially influenced gaming going forward

I could also make a case for The Legend of Zelda with how it blended so many elements together for such a new experience: Arcade-style gameplay mixed with open-ended design and intelligent item use for progression. Also, saving on cartridge (disk cards in Japan).

This is a tough one. Nintendo has been there at so many important junctures in the growth of the industry.

I think the top four are Donkey Kong, Super Mario Bros., The Legend of Zelda, and Super Mario 64.

I guess I’d give it to Super Mario Bros., for codifying a new type of gameplay experience, for making Nintendo and Mario household names, and for changing the fortunes of the North American video game market.

However, I don’t think Super Mario 64 is far behind, if only because the change it introduced was so sudden, complete, and definitive. At a time when console developers were taking baby steps into 3D, Nintendo EAD knocked it out of the park on its very first try, and in the process set a template for non-linear level design, camera movement, and analog control in a 3D space.

Super Mario Bros. Everything about it is innovative. The way music affects the experience. Design of characters to circumvent graphical/storage limitations. Incredible response to controls (40 years later there are side scrollers that do NOT play as tightly). But, mainly, playing for an objective that has some story behind it, not only the loop we had for most Atari-Odissey years. Doing all that why being incredibly FUN when most gameplay experiences were not really so is a gargantuan task, in which they succeeded.

Donkey Kong Arcade - it established what platformer mechanics are, and although some of Nintendo's later platformer games actually ruined the genre from my POV (albeit bringing in more people to it), DK Arcade was pivotal in influencing exploration platforming subgenre that led to Metroidvania and puzzle platformers.

Super Mario Bros. It's the game that gave birth to an industry... an industry we're still playing in.

Zelda, open world adventure that set a standard that was so ridiculously ahead of its time.

Probably Super Mario Bros. followed by Super Mario 64.