Darn it, the 20 year cut-off at 1995 eliminates a couple of my favourites. But with that being said, let's take a look. Expect this to change a lot as I think of more awesome games.
1. CHRONO TRIGGER (1995, SNES)
This is probably one of the best JRPGs ever made. It has a simple-to-understand yet compelling story, with multiple endings and New Game+ both adding a lot of replay value to the game. It has aged quite well since its release in 1995. Seriously, if you haven't played it... play it. And if you have played it... play it again.
2. KIRBY AIR RIDE (2003, Gamecube)
Oh the hundreds of hours I've spent playing this game. I'd literally play this and Smash Bros. Melee all summer back when I was a wee lad. Unlike Smash Bros. Melee, I still enjoy playing this game every now and then. It's a racing game that uses the analog stick and the A, L, or R button. If you wanted to, you could play it using only your left hand controlling the analog stick and the L button.
3. DON'T STARVE (2012, PC; 2014, PS4; 2015, Wii U)
I've blabbed about this game a lot. Cool art style, dark sense of humour, no mercy, tons of fun (especially with friends), easy controls, relentlessly difficult, highly addictive, and humourous with PC mods. Unfortunately, the bulk of my time spent playing the multiplayer version is tainted with memories of my ex. But, if the interest in a playdate is high enough, then I'll be able to replace those memories with good ones spent with you guys. Heck, I've made it farther on my own than I have playing with her.
4. SUPER SMASH BROS. 4 (2014, 3DS/Wii U)
The controls in this Smash Bros. game feel just right. The characters aren't floaty (right, Brawl?), but they don't fall to the ground like stones (looking at you, Melee). It's much more satisfying when you score a hit on someone, and the characters are much more balanced than in previous installments. The game relies less on exploiting glitches (*cough* Melee) and more on skill to succeed. Bulky characters are scarier, since the new "rage" mechanic allows their weight to play to their advantage by boosting their strength and can even the odds when you are Ganondorf at >150% damage. The DLC, while expensive, will add new life to the game for quite some time.
5. SIMCITY 4 (2003, PC)
I like building stuff. This game is why I like building stuff. Same with Simcity 2000, but Simcity 4 looks pretty and a thriving custom content community. Simcity 2000 has aged about as well as floppy disks and dial-up internet.
6. HALO 4 (2012, X360)
It's about time Halo showed up on this list, eh? This game had shiny graphics for a 360 Halo game, and launched my forge career. If it wasn't for this game, I wouldn't be clad in pink right now.
7. HALO 2 (2004, Xbox)
I have fond memories of this game from when I was a wee lad. I remember mashing up Flood with my brother and taking potshots on enemies with Beam Rifles on Delta Halo and Regret. It may not have aged well, but it has serious nostalgia value going for it.
8. MEGA MAN 8 (2000, PSX)
Very fun game, and for a Mega Man game, it has aged very well. The variety of upgrades you can get adds quite a bit of replay value, since there are only 40 or so bolts you can spend on them scattered throughout each level. Once you spend them, they're gone for good! But the cutscenes are by far the biggest drawback and some of the worst cutscenes I've seen in a game. Seriously, Mega Man doesn't sound like a man at all and Dr. Light sounds like Elmer Fudd.
9. FINAL FANTASY VII (1997, PSX)
The story and the gameplay have withstood the test of time. Its legacy is legendary, and as such is frequently regarded as the best Final Fantasy game ever released (alongside Final Fantasy VI). It is the first Final Fantasy to have pre-rendered cutscenes {and these cutscenes don't have awful voices like Mega Man 8}. However, its graphics have not aged well at all. The 3D models look really clunky and just awful. If bad graphics don't bother you, then give this game a go.
10. LEGEND OF ZELDA: MAJORA'S MASK (2000, N64)
This is the darkest Zelda game made until Twilight Princess came along. It has this sense of dread literally looming over your head at all times with the moon itself giving you the stink-eye whenever you look up to the sky. There are a variety of sidequests which contain the most memorable moments. However, these are tainted with the depressing fact that you have to reset time quite often after you complete these sidequests. You will remember the accomplishments, but the people you helped will revert to their troubled selves.
11. LEGEND OF ZELDA: SKYWARD SWORD (2011, Wii)
Dat story tho. Zelda is not just some princess you have to save this time around; she's your best friend. That incentive alone makes it my favourite Zelda game in the entire series. You have that extra motive to save her. Picture it like this: would you rather save the prime minister, or save your best friend? I'd save my best friend because I share memories with them, whereas the prime minister just makes me lose faith in humanity. And the music is also pretty awesome. It has solid gameplay, and some cool bosses. But I digress... THE STORY IS WORTH IT.
12. SUPER MARIO 3D WORLD (2013, Wii U)
It's a 4-player Mario platformer that doesn't stink like New Super Mario Bros. Wii or Wii U with a terrific soundtrack and cool powerups. It addresses many of the problems that the aforementioned games have with their multiplayer modes by making players share a combined pool of lives and giving everyone access to a stockpile of spare items. Also, you don't respawn as a little guy, and can emerge from your respawn bubble when you have some safe ground underneath you regardless of whether or not another player frees you. Each character has their own strengths and weaknesses, which accent each other nicely in multiplayer sessions. And might I add that it's 3D? No more 2D shenanigans!
13. SUPER MARIO GALAXY (2007, Wii)
It took 3D Mario games to the next level: space! And not only did it take Mario to space, but it brought a great soundtrack, solid gameplay, and planetary gravity into the mix! It made battles against Bowser not just a struggle to rescue Peach, but it made them a struggle to save the entire galaxy. The soundtrack really contributes to the atmosphere with its orchestrated sound. What more could you want?! Hm...
GOLDEN OLDIES (released before the 20 year cutoff of 1995)
1. Super Mario Bros. 3 (1988, NES)
2. Final Fantasy VI (1994, SNES)
3. Final Fantasy I (1987, NES)
4. Sonic the Hedgehog 2 (1992, Genesis)
5. Ninja Gaiden (1989, NES)
6. Sonic the Hedgehog 3 (1994, Genesis)