Interesting picks. Though it dawns on me I probably should have also done a list based on how game's personally affected me.
My OP list was simply based on how fun I found them to be as games, and how strong a narrative they may have had. In other words, trying to take personal bias of "It did this for me, therefore I think it's one of the best games".

Anyways, list if the criteria is how it affected me personally.

10. Phantasy Star Universe
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This was the first Console MMO I ever touched that I stayed on for more than a few minutes. 
It's a word where I oddly have little words to say, but a lot of feeling as to why it's good. It just all played... right.
You had your player home to customize, your appearance was never dictated by your gear or loot drops.
It was very user friendly so meeting friends, socializing, making groups etc. was incredibly easy.
It had no kind of grinding feature that took you away from the game, while most MMO's you may go "I need to do this for a few hours for this loot" you don't get that.
There is no easy sink, your fastest way to level and progress is simply to play the game naturally over time. And all the dungeons have low level and high level versions, so you never had to grind to X level before being to do Y mission.

The only real issue I have is the dungeons never seemed to encourage any strong tactics or inner party play, it was fairly easy to just stream through. But then again, that may have been on purpose. Cause if they truly were that challenging you'd get tons of players waiting around for 30+ minutes waiting for the right class mix up to show up... and that wouldn't of ended well.
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9. Spore
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Now, this was a game where my initial playthrough is "OMG this is so cool!" but every future one is "Meh, eh... This kinda sucks".

It became a huge let down at space stage, hell creature stage is my favourite so I drag that shit on as long as possible. What puts it on the list is quite simply it gave me something new. It let me customize so many things, I could let my imagination pretty much run wild. And I know the obvious answer to this is going to be "Minecraft!", which is true. But Spore also gave me a feeling of leading an entire race and civilization while customizing everything in it. Minecraft doesn't give me that sense of power, I'm simply a super steve who punches tree's and makes giant forts.
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8. Fallout 3
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My first real open world experience that could really keep me engaged. I loved the theme, the wasteland, the guns, the plot.

This was a game I sank so many hours into, it basically was my version of Skyrim as a kid.
Now as I got older New Vegas came out, and I personally find it to have a better world and mechanics (hence in OP list it's ranked higher), but it was Fallout 3 that first threw me into that type of game and gave me a real taste for it.
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7. Dark Souls
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Challenge!

Anger, pain and challenge! Dark Souls was the first game I played that wasn't simply hard for the sake of being hard. It was hard but fair, and at the same time provided some realism so shit isn't 100% fair. Such as magic is ultimately going to be more powerful than a sword with the right training, players with more initiative and cleverness will get better items than you earlier, and their hard work is reflected in that gear. But ever change those players got, you get too so it's not a one sided system for veterans.

It's a game where once you know what you're doing it's not that much harder than any other game... But if you're not that sure... Then you're in shit, and the game doesn't show mercy. It was the first legit challenging game I touched (Other than XCOM) that actually felt fair and reasonable about it.
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6. Wolfenstein 3D
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This one's damn easy, it was my first game. Ever.
That's it, as a five year old my Dad thought it was a good idea to have me shooting Nazis.
So I was introduced to video games, as a five year old, shooting Nazis.
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5. XCOM Enemy Unknown
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This was my true first taste into a challenging game, but in a different sense than Dark Souls. 
Dark Souls was always hard at the moment, but XCOM was always fairly easy and forgiving... Until your mistakes add up...

Then it takes off it's innocent front and delivers alien cruelty onto you.
Made a mis-call with where to place your guy? Dead.
Didn't focus on weapon research... Well you couldn't kill that Beserker in time.
Focused on too few soldiers? Oh look, here's a bad mission, enjoy your campaign reset.

Needless to say, this game is meant to be played with Ironman on. 
I just wish it wasn't so cruel as to warrant say a complete restart after one really bad mission.
Which, the Long War mod helps address... but it makes up for it with far more challenges.
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4. The Walking Dead Series
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Quite simply it's a narrative masterpiece.
The ending of season 2 truly sent it home, this is a game that was able to engage me completely on it's story.
And it's almost complete lack of gameplay never felt like a burden.
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3. Halo 3
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It's story, passable, but enjoyable.
It's Multiplayer? Custom Games + Forge = Social Godsend.

This was the game that got me exposed to the online community, that turned my gaming hobby from being a solo hobby to a group hobby.
If not for this game, I wouldn't have been nearly as eager to enter the Xbox LIVE community, and ultimately the gaming online community as a whole.
And to this day it carries great memories that I joke about with friends who were there.
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2. Kotor 1
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My first exposure to a video game that held a strong compelling Narrative. 

This was the game that turned my gaming hobby from a minor pass time into almost a full-time investment.
It's also the game I attribute to me being such a big Star Wars fan, the movies were always good. But the narrative? Eh.

Kotor is what really grabbed me and dragged me into the Star Wars Universe. Really got me invested in the lore, and got me invested into Games.
Because now games weren't just a simple pass time, games had proven themselves to be narrative masterpieces.

Fun Fact: It was my strong interest in Star Wars that start a chaining of events ultimately leading me to the RP Guild. If not for Kotor I may have never joined the Guild.
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1. Mass Effect Trilogy
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This was a game I picked up on the fly as a 12 year old, because it was advertised to me as a mix of my two favourite games at the time. Halo and Kotor.
What I got far surpassed them both.

Although today I admit Walking Dead has an even stronger narrative, it was Mass Effects narrative that drew me in to such an extreme point that Kotor didn't compare.
If Kotor is what got me heavily invested in gaming, Mass Effect was what kept me stuck to gaming... for life.

It is also here because the trilogy slowly released from when I was 12 to 18. It was the my #1 video game for all of my teenage years basically, never had a competitor until Walking Dead.
Mass Effect, and Mass Effect alone is what stood as the prime example of gaming and narrative to me growing up. In that sense, it has impacted me more than any other game has ever done.
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