1. Donkey Kong (ARC) - first game I can definitively remember playing
2. Legend of Zelda (NES) - the Zelda series has had a profound effect on my gaming life
3. Ultima III: Exodus (NES/PC) - cultivated my love of RPGs
4. Final Fantasy VII (PS1) - this game was a major cultural touchstone. Was the first game that convinced me to get a non-Nintendo system
5. SimCity (PC) I spent a lot of hours with the original.

Space Invaders (1978)
Pac Man (1980)
Frogger (1981)
King's Quest (1984)
Jet Set Willy 2 (1985)
Hyper Rally (1985)

My tastes changed of the years though. I got very much into rts and tycoon games in the 90s starting with Civilization alongside adventure games, rpgs and mmorpgs in the early naughties starting with FF7 and Everquest in the late 90s, more into fps and 3rd person action adventure games around the 2010s, simulation and sandbox games after that, currently more into puzzling and rhythm games in VR.

Pleasantly surprised to see SimCity being mentioned so much.

I argued with myself whether to put it or not, but I guess I fell out of the genre by late 90s/early 00s. That said, it was the first game that I've played that was more sort of a game and more like a project with no goals except the one's you've set yourself, so I'll have to reedit my post and put it there as honorable mention (along with few others).

Ff7,8,9
Diablo(ps1) and diablo2
Warhammer dark omen

Tekken3 and dead or alive 2

Starsiege tribes ( damn that pc game was cool)

Might and magic 6,7,8 

Age of empires 2

Kings field(2)

Breath of fire 3

Ratchet and clank
Unreal tournament
Ffx
Quake3
Devil may cry
C&C red alert 1 and2
Starcraft

Those shaped my childhood's taste in games mostly because my dad bought them,thx dad for making my gaming childhood so freaking awesome.

There are probably as large of a pool of games that shaped my gaming taste by not being my cup of tea. Going to mention a few of them as well. 

Doom - A classic for a reason. This was my first introduction to the FPS genre. I have visited a few games a few times after. Counter-Strike, Unreal tournament and Halo to name a few. Just seeing a gun, shooting anything in sight from the first person perspective felt like I was doing the killing, and I did not like that one bit. When I play video games I do not put myself in the position of the one doing all the things in the game, I view it more as enacting someone else actions rather than my own. The level of violence put me off a lot as well making me skip out on games I later figured I enjoyed.  

Fire Emblem Sacred Stones - I loved the first GBA game. I loved the Japanese only first GBA title in the series as well. I come to despise the Sacred Stones. what I liked the most with the first game was building my characters. The number of items, units and experience points that could be gathered from one playthrough was limited. What you got is all you get and managing that was one of the things I enjoyed the most in Fire Emblem. I know I'm very much an outlier here but being able to farm experience on unites that constantly respawn is my least favorite aspect of any game. And no, just not doing it does not work for me, that it is even possible remove all pride from building a character in a game for me. I just have no value if it can be done infinitely, then the deciding factor is time spent, not skill.  

Backpacker 3 - I do not think this series exists outside of Scandinavia. The first two games made learning cool and fun. The third installment made it clear that the learning part was not the fun part. Everything I liked about the first game was removed all that was left was questions. One of my favorite games ever is an educational game. I have not tried one out since the complete trash Backpacker 3 killed the series and the genre. 

Was thinking about SimCity as well. SimCity 2000 to be exact. But even if I really like that game, it did not have a meaningful impact on my gaming taste going forward. 

I find its influence on me to be primarily as first "software toy" that I've dabbled with, not so much as strictly intro to city builder genre. I mean, I still occasionally have desire to get something like Cities Skylines (but I don't, since I would be lost to my family for at least a month), but I can see some of SimCity DNA in games I've played since, like various Tycoons, or even Frostpunk or KSP. That's why I've put it in honorable mentions (since, on overall, it's not as heavy of an influence for me as other games I've listed).

Of course, that's what this thread is about, games that shaped us, which is expected to be very different due to different eras (and regions) we grew up at, different gaming systems we were exposed to and, ultimately, different tastes. I.e., while I always recognized SMB as great titles, I was quite lukewarm about them, since that's not were I wanted platforming to go, and why I found exploration based platformer genres much more appealing to me - hence my formative experience was not SMB, but proto-Metroidvania from C64 that came before it.

Three clear ones that are also my to this day favorite games:

1. Phantasy Star: my first JRPG... of MANY

2. Wing Commander: my absolute favorite series, has couple basic ideas I still hope from games: wingmen - the general concept that it's not you against 100000 enemies but there's also others on your side - and branching that allows you to keep moving on even when you fail/lose something

3. Secret of Monkey Island: 90's Lucas adventures and similar point'n'click may be my favorite genre

Beyond that it gets harder to choose.

4/5. Populous and Civilization certainly put me on strategy games for decades on.

6. My love for racing games has developed gradually, but surely Hang-On - Master System build-in and so one of my first games - had a big part.

7. Those were all in my formative years in 90's. Probably the biggest influence since has been Max Payne that really started my shooter era.

8. Special mention to Rachet & Clank 2: over two decades I had been PC gamer (aside plenty of retro emulation) and only on last decade I got an old PS2. I've never had much love for platformers/action-adventure, but this forced me to reconsider my opinions.

• Pokémon
• Tales of Symphonia
• Warcraft 3 (small shoutout to Age of Mythology since I played that first)

I don't think anything else comes close actually. And yeah the second two of those are my all-time favorites. If I had to throw in a few other honorable mentions it might be World of Warcraft, Smash Bros., the Witcher and Persona 5, mostly cause they were such huge experiences to me.

Legend of Zelda - love the series to this day

Metroid - while I have skipped the many handheld variations I still love this series

Final Fantasy - I played all of them including a couple handheld titles up to X. I tried to get into X-2 but just couldn't and this is where I put the series down

Goldeneye 007 - it was the first shooter I really enjoyed

Tekken - while I have never branched out from the series to other fighters (outside of playing various ones in the arcades) I loved this series up until monetization ruined the last couple entries for me

Splatoon - I had played many shooters including COD titles but would get burned out within a couple titles. Splatoon for me refreshed the 3rd person shooter genre with its fresh take and mix of platforming