This is a toughie OP, but I'll try to limit myself to five games (with some cheating) that are special to due to attached memories and such, in chronological order from old to new:
1) Super Mario Bros. / The Legend of Zelda - I list both because they represent the definitive duology that made me conscious of being a game enthusiast after I got my NES (my second console after having an Atari 2600 first), the former because of its instant joy and satisfaction and the latter because of how it set its adventure hooks in you for hours on end. Both games were very different but also equally fantastic as experiences that everyone - even those that never played games normally - wanted to try. Zelda in particular opened my eyes to the power of games as something more than round or score based experiences, i.e. as adventures that could last many a play session. The fact that my mom to this day remains a Zelda and action adventure game fan because of our shared experiences with the original is very valuable on a personal level as well. Mario also stood out because of how it helped pretty much all kids in my neighbourhood become fans of games, but also because it gave them a shared cultural experience (akin to Pac Man before it) to look back to, an experience that for many of us still lasts to this day.
2) Secret of Mana - While I like Chrono Trigger, FFVI and Link to the Past equally as solo experiences, this SNES RPG wins its spot here by being the game I have the fondest memories of while playing it in co-op with others like my mom, my high school best friend and finally my then girlfriend now wife. Its incredibly charming look, touching and memorable music as well as the timing based, beat 'm up style combat made it an instant classic for me and a game I think back to fondly quite often as a representation of more carefree times where hours would waste away playing it. I only wish its sequel could have released outside Japan in the day as I still never played that one really despite feeling it might rival this game.
3) Street Fighter II - Probably the game that blew me away the most out of any game I had seen before (since the Atari and NES days) with its pure awesomeness and instant playability. I never really encountered it - or any other fighting game before it for that matter - in arcades (due to them not really being a thing in The Netherlands) but the SNES version was enough to instantly fall in love with the game and the fighting game genre as well. The colorful cast, big sprites, music, special moves and the travel the world theme greatly helped with my enjoyment as I spent hours and hours fighting my friends or passing the controller with my parents as we tried to beat the CPU. While SFIII and V might have surpassed II for me in some ways, II definitely deserves the spot in this list.
4) Outrun / Outrun 2(006) - Not a series I played the most of in terms of hours (partly because not having arcades around meant I played 2006 first and had to be creative in playing the original in Shenmue or through emulation/ports), but one that has come to represent an actual emotional state for me, i.e. the jittery feeling of excitement and relaxation when summer rolls around again and the potential for road trips or things like laying on your back on meadows watching clouds drift by increases. Whenever I go on holiday, think back to my childhood summer vacations or just feel relaxed and excited to be alive I describe this as an Outrun feeling because of how my mind instantly goes to this series and how well it represents such positiveness and excitement. What I like so much about Outrun is the colorful, real life inspired locales you drive through that instantly remind me of movies like Canonball Run etc. The blue skies, sea and palm trees of the first stages help greatly as well, as does the music of course. One of my favorite recent holiday memories was finding my first Outrun cabinet ever in the wild hidden away in a casino in Las Vegas - it was the perfect blend of gaming nostalgia and actual holiday jitters! (Now bring Outrun 3 Sega / Suzuki-san!)
5) Demon's Souls / Dark Souls - I'm listing both as I like them equally (even though I did start with DeS before it became popular) and because DS proved that DeS was not a fluke but something FROM could actually recreate technically and financially within the modern games climate. I'd say that these games (as well as their sequels) pretty much single handedly keep my youthful obsessed form of interest in games alive with their large worlds, hidden lore, great combat mechanics and hours and hours of careful exploration based gameplay. In many ways they manage to bring to life what I dreamt of in my childhood when I saw illustrations for games like Zelda. The idea of a gaming climate without this series frightens me as nothing really comes close to what these games offer in many ways. I can't wait to see what Miyazaki and his teams cook up next.
Honorable mentions that - depending on when you ask me - could take a spot in the top 5:
X) Ico / SotC - Without a doubt two of the games that hit me hardest emotionally because of the sombre atmosphere and sparse looking worlds they bring to life. They come the closest to being a game version of emotional / sombre adventure movies from my childhood like like The Neverending Story or The Last Unicorn and for that alone they hold a special place in my heart. I consider them landmarks in game aesthetics that continue to inspire me and many other people as well (such as FROM's Miyazaki, who decided to change career to game development because of Ico!). Let's hope The Last Guardian will equal these two and become another instant classic.
X) Final Fantasy VII - The game I envied the most as a N64 loyalist at the time and one that eventually made me buy a Playstation as well. Due to my age and relative inexperience with JRPG's (outside of some SNES classics) and this game's fantastic presentation, music and character designs, FFVII became a landmark game for me that has come to define the PSX for me as a well as providing me with the perfect escape during the time in my life when my parents got divorced (as it was my dad who bought the PSX for me to use at his new place). I can't wait to see how the remake will turn out.
X) Sonic the Hedgehog (classic series, but mostly 1) - A game (series) I simply adore because of how fantastic it looks and sounds. The perpetual Other to what I was familiar with on the Nintendo consoles I did own as a child, and a game (series) that ever remained slightly out of reach / alien for me whenever I saw it being advertised on TV or visited a friend who did own a Mega Drive. Shinobi, Altered Beast, Golden Axe and Streets of Rage also had this effect on me but they don't stand out as aesthetically as classic Sonic does in my view.
X) Mega Man X - The game the finally made the Mega Man series click with me and the first one I beat (multiple times) as a child. I've loved this game and the overall series ever since.
X) Doom / Doom 2 - The games that defined PC gaming for me for years and the first FPS games I probably played. Still unequaled in some ways.
X) Super Castlevania IV - The soundtrack alone was enough to make me a permanent Castlevania fan as well as a permanent appreciator of game OST's. Just like with Mega Man X, this slightly more modern take on the formula made me appreciate the first three games in the series far more as well. A shame we might never see another game in the series again...(and yes, I'm still bitter Simon was not added to Smash 4...)