This list is largely for seasoned gamers those who struggle to find a game that feels novel, or stands out in quality. Tried to include a variety, and avoided making this into a "top ten" list. Hope this is helpful!
Stardew Valley In short, this is largely Animal Crossing meets Minecraft- but is very distinctive. One of the few "grindy" games I can truly suggest after spending thousands of hours playing video games- made by one person, this has character and can be played many ways, making daily tasks somewhat flexible.
Forza Motorsport 4 Arguably the best racing sim ever made, IMO. Obscene selection of cars, tracks, customization, and more. Other than multiplayer, there is little reason to touch a newer racing sim. A good example of how modern day gaming has declined.
EarthBound Truly a unique RPG with interesting humor, outstanding music and visuals, and tons of unexpected twists. This game is unlike any other. Some aspects have aged poorly, it can be grindy at times, and a guide is virtually mandatory- but I still highly recommend giving this a shot. If you do, I suggest the guide on starmen.net.
Kid Icarus Uprising* Uncomfortable to play with adult-sized hands.* Unique story and gameplay, with self-adjustable difficulty and decent weapon selection. Several massive story twists. One of the few games that truly capitalizes on all of the 3DS's capabilities. If you are interested, be sure to dodge spoilers.
Kirby's Adventure A somewhat simple but amusing and well-designed platformer. If you want a game that's easy to pick up and play but can offer some challenge, give it a try. Was re-released a few times.
Gadget Racers PS2 A fantasy racing game that maximizes customization on every level, and was one of the first to incorporate flight and swimming cars- and somehow does so better than modern Mario Kart. Racing is simple but provides ways to improve and get an edge over the competition. One of my most played games! Physical copies are cheap.
Super Street Fighter 4 For casual gameplay, I think it's overall the best blend between new and old school mechanics- fun single player mode. The 3DS version allows for special moves to be hotkeyed to the touch screen if you find some difficult to execute and want a more chill experience.
Melee An extremely raw and unforgiving fighting game that challenges the human limit, virtually endless room for improvement. Although playing competitively I typically wouldn't recommend, playing casually is certainly an experience. Maybe check out community projects, too.
Crash Tag Team Racing* Lots of esoteric and childish humor.* The most controversial game on this list, CTTR is a mix of racing and platforming. The end result is not the best at either- but does a great job at holding your attention through various objectives, scenery, and absurd zaniness. Vehicles 'clash' together forming one mega car with a racer and a gunner. With some notable VAs and questionable lines, you'll definitely raise an eyebrow.
Metal Gear Solid: Snake Eater MGS is one of the few stealth series that truly nails it, but Snake Eater stands out for its powerful story, varied gameplay, and genuinely outstanding cutscenes. Controls are 3rd person and are slightly more primitive than newer MGS games, but this gives it a special characterization.
Noteworthy mentions
Ecco the Dolphin A fairly unique 2D platformer that's set underwater- for those who want a very great challenge. Its extreme difficulty and sometimes unclear tasks has led to controversy, but if that's what you want, this game is certainly an experience. One of Sega's more historically important games.
Duck Hunt Similar games can be found in arcades, but the NES's Duck Hunt is the OG way to experience simulated aim and shoot. Requires a CRT TV...
Pushmo Walk around as a cute little sumo wrestler, pulling in and out blocks to reach the goal. Paired with the 3DS's 3D, this game makes great use of a puzzle concept that would otherwise be difficult to pull off. Unfortunately, due to the closure of the eShop, there is no way to purchase the game.