Big Games I've never played:
Any Call of Duty game. I prefer my FPS to be sci-fi.
Mass Effect: No excuse for this one. I've just never played them.
Gears of war: Found it incredibly boring and too dudebroish.
Demon's Souls/Dark Souls/Bloodbourne
Fire Emblem: I have an Ambassador Fire Emblem game but I haven't touched it.
World of Warcraft/Destiny/any MMO: MMOs are not my thing.
Dark Souls and related:

Big Games I couldn't get into:
Any Sonic game past Sonic 2
KOTOR: I just didn't like the combat system
Final Fantasy XII: The style, atmosphere, and combat were so different from other Final Fantasy games that I was turned me off. I don't mind different, I just didn't like this kind of different.
Lumines: At first I liked it but it quickly grew boring for me.
NInja Gaiden (Reboot): I found the story to not be interesting. I prefer the NES games.
Harvest Moon: This game is jut terrible.
Monster Hunter: I tried the 3DS demo and hated it.
Dragon Warrior/Quest: I prefer Final Fantasy.
Super Mario Land 2: 6 Golden Coins: I just don't like this game like everyone else does.

I'll add more as they come to me or if you add something I forgot.

Anything in blue was added after the original post because I either remembered or your post inspired me.

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@Pokefreak911

I try all the time. If I could find Rainbow Curse for cheap I'd probably pick that up.

@DrFinkelstein
I haven't played many of the modern games actually, which I should change as Triple Deluxe and Return to Dreamland both look fantastic.

Oh, I might as well add in that I have never played an MMORPG in my life. No Everquest, no WOW, no... whatever other ones exist.

And I most likely never will play one. Just don't have any interest in them.

Zero said:

Oh, I might as well add in that I have never played an MMORPG in my life. No Everquest, no WOW, no... whatever other ones exist.

And I most likely never will play one. Just don't have any interest in them.

Ditto

@Pokefreak911
Sorry I only played one Kirby game and that's the first one. I've also never played a Wario game or Animal Crossing.
I also never got into Pikmin despite loving other RTS games.

Okay everyone get you guns out because I about to say heresy.

I don't seem to enjoy any Mario games at the moment outside Mario Kart and Super Smash Bros.