A lot of "infamously terrible" games really aren't as bad as people say. I kind of like Bubsy 3D, for instance; I like the mostly flat-shaded graphics and the level designs. Superman 64 is not a good game, but the flying-through-rings half can be fun once you get used to the controls, which are not THAT terrible; it's the other half of the game that I don't like. Carmageddon 64, the other "worst" N64 game, I actually think is kind of good. It's no match for the original Carmageddon for PC, but it's not the worst thing ever either. Etc.
However, there are some legitimately atrocious games out there. So sure, why don't I mention the worst ones I own? Starting with the worst, here are my top two...
1. Billy the Wizard: Rocket Broomstick Racing (Wii) - I like the Wii, and I like motion controls, but the controls in this game are totally, hopelessly broken. Just TRY to actually fly around the courses in this flight-racing game, it's nearly impossible thanks to the busted, and motion-only control scheme! Outside of the unbelievably bad controls this is just another super low budget port of a European PS2 game, but thanks to its controls I don't think I have ever played a worse game.
2. Shadow: War of Succession (3DO) - On the note of bad controls, this game is STUNNINGLY terrible. People like to talk about how bad Way of the Warrior for the 3DO is, or any version of Rise of the Robots, but those games play like Street Fighter II or Mortal Kombat compared to this unrivaled disaster. I got this in a bundle knowing that it's supposed to be horrendous, but even that did not prepare me for just how unplayably abysmal the controls in this game is! Shadow is a Mortal Kombat clone, and it looks like they had some budget for a CGI intro and the expected MK-style digitized-actor characters, but the controls barely even exist, characters have almost no animation or moves, the games' speed is way off, and so much more. I have never seen a game that shipped so completely broken from the days before you could patch things.
And some other games that are really terrible: Supreme Warrior (SCD, 32XCD, etc.; this is a complete mess of a FMV fighting game, don't play it) and Hardcore 4x4 (Saturn; this is okay if badly dated on PS1, but the Saturn version runs shockingly terribly, what is a framerate?). Also really bad, but not quite as terrible as those, are Largo Winch (PS1), a terrible stealth action game, and Beast Wars: Transformers (PC/PS1), a really bad 3d action game. The Famicom Transformers game is awful, but I at least think this one is even worse.
I also want to mention a couple of games that aren't really among the worst ever but I greatly dislike for other reasons; Metroid: Other M (Wii; for what it did to Samus), and Gauntlet: Seven Sorrows (PS2/Xbox) for not playing like a Gauntlet game, being bad and boring, and maybe ruining what was a favorite franchise of mine.