These are ranked from all-time-favorite to still-worth-playing but-only-if-you-really-like-these-kind-of-games. Note that I take my time on acquiring gaming hardware until it's very long in the tooth (and super cheap!!) so although this list may be written in 2017, none of these games were released even remotely close to that. The newest might be 10 years old??
Legend of Zelda, Ocarina of time (N64)
Nostalgia plays a role here, but this game is still amazing. The world is huge and exploring it feels like a real adventure where you get to decide where to go next. The slow evolution of your character's abilities, though trite (especially today!), makes it feel like you are growing with the game. Combat is only moderately challenging, but never devolves into a button masher. The dungeons are full of puzzles, and are at least moderately varied. The controls are perfect. Graphically, this game is extremely dated, but the game play is not. And the graphics are not so bad as to get in the way of the game, though it probably helps that I first played this when it was relatively new. There are many remakes on recent Nintendo platforms, or you can easily play with a free emulator on a PC. Though the N64 controller was pretty different from the dualshock design that's pretty much swept the console/pc world, so it can be hard to find a elegant control scheme.
Ratchet & Clank (PS2)
This is a great series, but the first game is arguably the best. It's a "puzzle shooter", in that there are a huge number of guns that each behave in distinct ways. Half the challenge is deciding which would be best for the current situation. It's also a first-class platformer with lots of running, jumping, and exploring. Each level typically has several paths to explore, though to finish the game you will have to play them all, so it's not as open as a zelda game, but the design does a good job of hiding any linearity. Much like zelda your character gets more and more powerful, meaning that you can go back to earlier levels to complete challenges that were impossible before. There is a tiny bit of hidden stuff, but for the most part there are no secrets. Just blast your way through the levels and when everything is dead you win. Graphically, the game still shines; with a cartoons sci fi aesthetic. The game play is also up to modern standards, though you will find yourself repeating parts of levels more than you might like because you didn't get far enough to reach the continue point.
Mario 64 (N64)Nostalgia may also play a role in this pick, given its age. Certainty, as the first open-world 3D platformer, it deserves to be on this list for historical reasons. And frankly, it's hard to imagine that you haven't played it, so maybe I shouldn't belabor the point much. But part of the reason to go into so much detail on the first few games is to help define what this list is about so I will spill at least a few characters. Mario 64 succeeds for several reasons. Unlike the earlier Mario games, each level has a series of challenges to complete, keeping gameplay more varied; there's a level select stage, so it's ease to jump (heh) around to different levels when you get stuck; most of the levels are fairly open and exploration oriented, rather than linear. Keeping with the Mario tradition, there's lots of secrets to discover and all kinds of tricky jumps to execute, and combat plays very little role except for bosses. It really sets the standard for pretty much all mario games that have followed. I find the controls just slightly less smooth than in later games, but perhaps that's because I played on an emulator, using a dual-shock style controler. The graphics are very crude by any standard, but don't hold the game back much. If you haven't played it, I suggest that you do for historical interest - you'll be shocked how much of the mario style was established so long ago.
The rest...Having established the pattern of what I'm going for here, I'm going to go for more short and sweet descriptions of games; as always google for further details.
Jak and Daxter: the precursor legacy (PS2)
A really pretty "gather the magic orbs" kind of game, ala Spyro. Lots of platforming, and good level design.
Mario Sunshine (GameCube)
What if Mario was a FPS? Would he have a gun, or a power-washer backpack that is as good at cleaning away black sludge as it is giving short hover assists for tricky jumps? Well that sounds like a weird mashup, but the game is actually quite fun. There aren't that many open worlds to explore; instead each "overworld" also has a bunch of fairly linear "challange" levels which are kind of like 3D versions of NES mario. I feel like it was kind of short.
Zelda: The Windwaker (GameCube)
Ever felt like Zelda was too epic and didn't feature enough watersports? This one is for you. I've heard that it's only got about 10 hours of traditional zelda gaming, but I haven't played it recently so that might be an exaggeration. I do recall a lot of sailing between islands that was only fun the first 200 times. The islands themselves are traditional zelda overworlds with traditional dungeons. Since there aren't that many zelda's it's still worth playing but the bang for your buck is pitiful.
Mario Galaxy (Wii)
The gimmick here is that some levels are tiny planets that you can circumnavigate in a minute or two. It's large, it's open ended, and it's the mario you know and love from 64. For some strange reason I have fonder memories of the shooter mario (aka sunshine), though. But it's very solid
Spyro the dragon (PS1)
I'm not sure of the history of the "collect all the thingies" genre, but this is one of the best early examples. Every level has 400 gems. There are weak monsters wandering around that make it slightly hard to get all the gems, but mostly this game is about exploration, tricky jumps, and figuring out the path to platforms that cannot be reached directly It makes the most of a few very simple gameplay mechanics. I found it surprisingly sublime, and was tempted to put it higher on the list, but it is such a basic looking game with such repetitive gameplay that it found it's way down here. It looks much better on a PS2 with texture smoothing on.
Ratchet & Clank Going Commando (PS2)More Rachet & Clank. Slightly heavier emphasis on the run and gun 3rd-person shooter aspect of the game, and less on the platforming and exploration, but not really enough to change the flavor of the game.A great looking game and a solid choice if you liked the original.
Jak II (PS2)
Jak and Daxter made over to be a lot more like Ratchet and Clank. The guns are much fewer and less interesting than R&C. The collect the thingies bit from the first game is almost entirely gone; instead you just have to get to the end of each level (many of which are rather linear, but well disguised) . Way, way, too hard for its own good, with lots and lots of replaying each level because you died. I'm sure I died over 100 times on some levels. I would rate this one much higher if it didn't have that going on. It looks fantastic, and is a lot of fun until the difficulty level gets ramped up. I don't care too much about plot line typically, but man this one is hard to follow and there's a lot of it. Playing Jak and Daxter first helps, but I kind of think they meant for it to be confusing.
Tak and the power of the JuJu (PS2)A collect all the thingies game, but with more interesting puzzles than most, and pretty good graphics. Sort of a Jak and Daxter + Spyro + something with puzzles + "humor". Sometimes frustrating. Combat has a very strange place in this game - it can be hard, but you come back to life after dying without losing any progress at all. I think they could have skipped combat altogether or made it easier but given death some actual consequences. Sometimes I get lost in the levels because they are a bit visually repetitive.
Zelda Twilight princess (GameCube/wii)
Felt pretty linear for a Zelda. Otherwise sold, typical zelda style game.
Ratchet & Clank up your arsenal (PS2)
The continued evolution of R&C away from its platforming roots and towards being a 3rd person shooter. Still has a ton of interestingly different weapons, and still has plenty of platforming, so I still recommend it, but not as much as the 1st or even the 2nd of the series.
Starfox Adventure (GameCube).
Much more Zelda than Starfox. An odd game that got it's Starfox branding late in life. Somewhat linear, and a too much emphasis on combat, but still a fun Zelda-alike.
Spyro: Ripto's Rage (PS1)This game looks so much better than the first Spyro. Some of the levels approach PS2 level complexity, though certainly not ps2 level graphics. It might be worth seeing just as a testament to what a well-programed PS1 can do; I think it's easily one of the best looking 3D games for the system, esp. with PS2 texture smoothing turned on. Sadly, the sublime simplicity of "collect all the thingies" has been traded in for a bunch of varied challenges like animal herding or killing all the monsters. Though challenging they don't tend to be that fun. Reminds me of how Mario 64 introduced reusing each level 8 times, but requiring different tasks each time you return. The tasks are less fun but at least there are only 3-4 per level here. You can still collect all the gems, but there is no real in-game motivation plus they are not hidden as cleverly.
Games I haven't played but should probably be on this list
For being written in 2017 this list is horribly dated. Even I know of many games that should be on this list (and even own some of them), but haven't played them personally yet. Some are listed below. I'd be eager for suggestions for more in the comments section.
Zelda Skyward Sword
Zelda Majora's mask