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.
There are tons, but I think the glaring one is Mega Man. They're supremely well-designed games, but I associate them with an annoying kid who lived down the street from me, so consequently I never really played them. Tried to to get into them as an adult and just didn't have the patience. But that's on me, not on Mega Man. I can tell it's a good game series.
The big ones for me as a Nintendo fan: Pokemon - Bought Platinum, sort of liked it, but not enough to finish or ever buy another Pokemon. Too many random battles. Wario Land - Played a couple of them, couldn't get into them. Not my style.
Non-Nintendo: Grand Theft Auto. Played one once years ago (III I think?), didn't find it compelling, never bothered since. Gran Turismo. Bought GT3, tried to get into it, found it supremely boring.
I could say Call of Duty but I found it about as fun as any other FPS outside of the small handful that I actually love, which is to say it was ok but nothing I felt any need to play beyond the small bit I messed around with at my cousin's house.
But seriously, hm... I guess I'll say GTA like someone else mentioned. I can certainly see why they're popular, but not my thing. I guess this goes for any realistic-styled game in general, though I don't actually try to play that stuff.
Also, sports games of course. Madden for example. And fighters. Street Fighter, etc.
There might be more but nothing's coming to mind yet...
Pokemon - played two of the games (Blue, X) and just don't like them as much as some people do. GTA - The stories are decent at best, and the gameplay just gets dull after a while. Assassin's Creed - Parkour is fun, but the rest of the game is...eh. II was good. I didn't care for the rest of the series.
Skies of Arcadia - The battles are too drawn out. I see why people adore this game so much...I just don't.
Many, many non-Nintendo franchises. Not sure if it's BECAUSE I don't have other systems, or what, but most GTAs, Gran Turismos, Gearses of War, Call of Duties, etc etc don't really appeal to me.
As for Nintendo series, I LOVED Pokemon for the first two games, but its appeal has almost entirely diminished for me, unfortunately. I used to say Fire Emblem and Advance Wars here, but this gen has made me a believer in both.
Metroid is probably the biggest one for me. I tried Fusion, Prime, and the first one. I like collecting everything in platformers- all the Music Notes in Banjo Kazooie, all the Mega Man X health upgrades, all the upgrades in Kid Icarus- so seeing upgrades that I can't get until later stresses me out. This exists to a degree in X, but that game offered a level select, so when I had the appropriate powerup I could get to that item I couldn't reach before in less than 5 minutes. And it was easier to remember where those things were too. I also don't like being dropped into a spot in a game where I could explore four given areas and being told to pick one. More often than not 2 or 3 of those pathways are going to be temporary dead ends where I'll have to come back later, so it just feels like a waste of time. Maybe that's why Fusion appealed to me the most.
I'm pretty sure these games are good at what they do, but it just doesn't appeal to me.
Big ones that I've tried to love but just can't get into for one reason or another:
Smash Bros. – I just don't get it, I think it's funny and fun for a few hours but I don't get why people love it so much and are so obsessed. Pokemon – The game systems are too complex and impenetrable for new players. Monster Hunter – The game systems are too complex and impenetrable for new players. Also MH4 has doesn't look that great on the 3DS, it's too big of game to fit comfortably on that system. Maybe on the NX... Xenoblade Chronicles/modern Final Fantasy/JPRGs – I like action RPGs where you control one character (Zelda, Dark Souls) or turn-based RPGs (Dragon Quest). I don't like games where you have to control large parties in real time. Pikmin/Starcraft/any RTS – Some dysfunction in my brain keeps me from being able to control multiple characters simultaneously in real time. Mass Effect – Didn't love the shooting or the RPG systems. Assassin's Creed – I can never make it though the hours of restrictive and tedious tutorials to get to the open-world stuff which is the main draw. GTA (with the exception of Chinatown Wars) – insanely impressive and in-arguably uber high-quality masterpieces of game design, that I just get bored with. *shrugs* World of Warcraft/Destiny/any MMO – maybe I'm too antisocial? I just want to explore and have fun at my own pace!
My brain is immediately hostile towards any war shooter from the second I see it, even though I realize that is unfair. My favorite first person shooter of all time is still Doom, and these modern shooters seem to go in the opposite direction... slow paced, full of characters yapping at you throughout the entire game, restrictive in the way that they guide you and make attempts to be realistic, and just boring from a thematic perspective. I hate all that macho, patriotic, war glorification stuff in all mediums. I've played some Call of Duty multiplayer, and I understand the appeal, but it does nothing for me.
I'm worn out on pretty much all of the popular AAA games from third party studios right now, quite frankly, as they seem to be the same experience in slightly different packages... shooting feels identical whether it's third or first person, cover systems have lost their novelty, and regenerating health is a bastardization of the once unique recharging shield system of Halo. I'm done playing that game.
That's basically my opinion on the non-Nintendo games you mentioned. AAA gaming just doesn't appeal to me. I'm sure that I've missed out on some fun experiences but I don't feel hard-done by skipping them.
Final Fantasy: I don't have the patience to go through all that it takes to get through one of these games. But that's true of any RPG really.
For Nintendo properties: I simply do not understand the joy derived from Animal Crossing. I've watched people play it since its first iteration, I tend to fall asleep rather quickly. I've grown tired of Pokemon and Wario Ware. I've played a number of entries from both of these series and I just can't stomach them any more. The only Pokemon games I enjoy now are ones that use them as a skin: Pokemon Pinball; Puzzle League or current games like Shuffle.
I personally think that is half the point of the God of War games. I've really only played the 3 numbered games and I think Kratos being absolutely contemptible is what makes it interesting. Basically a villain fighting other villains in increasingly ambitious set pieces.
Grand Theft Auto - I find these games so incredibly tedious and just unfun to play. And this has been opinion since the first one. Pokemon - There is nothing in particular that I can point to but yet I just can't get into them. Dark Souls and related - The purest example of trying way too hard and way too much hype for such mediocre games. Ugly as sin as well. Mario Party/Golf/Tennis - Just does nothing for me even with all the Mario.
Final Fantasy is probably the biggest and most-heralded one that I've never gotten into at all. I may give FFXV a chance whenever that rolls around, we'll see. I'm more open to RPGs today than I was 10, 15 years ago.
On the Nintendo front, it's definitely Pokemon. My brother borrowed the first Game Boy one from a friend a long time ago, and I played maybe an hour or so of it and just could not get into it.
Pokemon is the big one on Nintendo systems. I've never played a Pokemon game, and I don't have any interest in trying one.
I've also never played a Fire Emblem game before, other than the demo for the 3DS version.
I own all of the Super Smash Bros. games besides the newest versions for Wii U and 3DS, and while I think they're pretty fun, I hardly ever play them and will probably never buy another game in the series.