It's called 'The Dungeon of the Board.'
It generates a mystical magical adventure for you from text.
Some bloke called 'Huinesoron' made it.
Whoever that is.

This is a series of flash games. Sure, they are flash, but this is what started the development of my mind. I used to play a lot of Flash games before 2010, so here is a list of some recommended sites. (Just Google this stuff)
Stickpage
Andkon Arcade
Nitrome
Eyezmaze
Armorgames
Hotairraccoon
Kongregate
Miniclip (Not really)
y8

Some particulars:
Epic Battle Fantasy, by Matt Roszak
Age of War, with music by Waterflame
Ultimate Flash Sonic
Death Vs. Monstars
My pet Protector
Get off my Lawn
Kids vs. Ice cream
Kids vs. Santa
Shellcore Command

I have a bunch of Sonic games on the Wii, due to the influence of a friend of mine. Sonic Riders: Zero Gravity was fun, but the music gets repetitive. Sonic and the Secret Rings is kinda bad, but that's only because I've played part mode waaay too much.

If you can somehow go back in time, I'd recommend LEGO Universe.

If you have a phone, I'd say go get Battlegems. Just saying.
I have a lot of other ideas, but they will just get more and more random.

-DCCCV, who just went down memory lane.

it's a life-changing, deep experience, it won Game of the Year in 2013, and many consider it to be one of the few games of this generation to be called Art. Best of all, it looks absolutely gorgeous, and the music is so soothing and you really can get into the shoes of your little avatar. It's a game about the journey of life and not the destination.

It's unique in so many ways - for example, if you play with an online connection, you could do multi-player, but only in the sense that you might by chance bump another player from anywhere around the world who's also playing the game at the same time. There is no text whatsoever except for the title and the credits, and no way to speak - you can only sing musical notes to gather energy and interact with stuff. Did I mention you have a magical scarf that you can lengthen to - get this - FLY? And best of all, there's a mysterious story to discover about the history of your race and who you are and what your destiny is, a story that fans still offer their own interpretations on.

It's not a kiddie game - it's a game with a deep message, a story that will urge you to return, stunning visuals, and an absolute relaxing effect that goes beyond any other game I've known about so far.

It's a single-player game and takes two hours at minimum to complete, but it is so worth it. You will want to play Journey more than once, I promise!

If you get the chance, I'd highly recommend the MOBA SMITE. Long story short, it's basically a mega-crossover crackfic of six different mythologies and it's ridiculously fun. I've heard this game is similar to LoL in terms of character playstyles and overall gameplay, but it's been growing into its own lately and I think it'd be worth giving it a look. Best of all, it's free! Just set up a Hi-Rez account and download the game from Steam and you should be good to go. :)

Speaking of Steam, I'd also suggest Undertale, because the story and gameplay are really well done, to say the least. Y'know what, I'll let Bogleech do all the talking with this one.

And another underrated Steam game I've played: Primal Carnage. It's Team Fortress 2 meets Jurassic Park, and they're working with more obscure dinosaur species, too. I haven't played it for a while, but I used to quite a bit during college and the experience was quite action-packed. Shame it hasn't gotten more exposure, they're doing quite a good job for a versus shooter with JP-style prehistoric creatures in it.

I'm a huge PC gamer myself but I'm not big on console games. Anyway, here's my list of PC games you may want to try out:

Team fortress 2 - While it is rated M, it's only because it's a first person shooter and in my opinion it should probably be rated teen. It's 100% free too so no problem if you don't enjoy it.

Minecraft - playing in survival mode may be time consuming, but there are tons of servers with many versatile custom minigames that you may enjoy. Practically anything is possible with this game.

League of Legends - I haven't played this game in a while but it's also completely free. Games take on average about 30-40 minutes, and there are so many different characters to play that it won't get boring anytime soon.

So let's begin with... er, between things already posted and your ban on RPG, it's already harder. I'll try it anyway.

For PC, you could try Starcraft 2, or find Red Alert 2 in order to usee its mod Mental Omega. or, you could go with the 20-hours long tutorial o XCOM Long War, Enemy Within. Long War can come later.

Aaand, if T-RPG are still allowed, you could try Fire Emblem Path of Radiance and Fire Emblem Radiant Dawn.

Oh, this is my jam.

Wii

Super Mario Galaxy 1 and 2. Platforming at its finest with a sublime orchestral OST to boot.

Donkey Kong Country Returns. Another excellent platformer with more sweet, groovy tunes. Made by Retro Studios, who also made...

The Metroid Prime Trilogy. First two on Gamecube and last one on Wii. A first-person exploration adventure with some shooter elements. OST and art direction are top notch. Thanks to the Wii's backwards compatibility you can play Prime and Echoes but you'll need a Gamecube controller and a memory card.

The Legend of Zelda: Skyward Sword. Yes, you need a widget thing for the Wiimote but trust me: the ride is worth it.


PC

Drop anything you're doing, get on Steam, and buy Bastion and Transistor. Do it. Do it now.

FTL: Faster Than Light if you want all the pain of Dark Souls and the fun of a spaceship management/combat simulator.

I'd suggest Half-Life 2 and its episodes but you said you're not into M games, right? Still, throwing it out there: it was and still is an influential FPS game.

And lastly, Minecraft because why not.

You might want to try WARFRAME. Aside from that, hm. Heroes of Might and Magic III is really good. I'd say Nox, but you said no RPGs.

but those are both Rated M...

...I think I speak for everyone when I state that Undertale is an excellent game (even if I have only watched the Steam Train playthrough)

... the various Lego games, which you can get for either PC or Wii. I have Lord of the Rings and Jurassic World, and have played Star Wars some too. Jurassic World seems to have aimed weirdly at children - in the levels based on the first film, no-one and nothing dies in gameplay or cutscene unless you mess up the quicktime-event raptor attacks in a single level - but LotR and SW are very fun.

Particularly LotR, since it's open-world across the entirety of Middle-earth, and after you finish the game you get to choose your character... yeah. :D

OH! And if you get it for anything other than the PC, you might be able to get the DLC packs, which include...

(Drumroll, please)

A mini-Balrog.

(It is purportedly possible to unlock this on the PC as well; the data certainly exists, but my copy refuses to open it up. If anyone happens to know how to do this, I would be very grateful.)

hS

I played several of them (made by Frogwares) and I love every single one I've played. Personal recommendations:

Sherlock Holmes: Nemesis, also known as Sherlock Holmes vs Arsene Lupin, from 2007. Sherlock is challenged by the French Master Thief, Lupin, to a game with the whole Britain's reputation being at stake.

Sherlock Holmes vs Jack the Ripper from 2009. Pretty self-explanatory methinks.

The Testament of Sherlock Holmes, also from 2009, set in the London of 1898, with Holmes presented as the main suspect in a case in which he is unable to prove his innocence.

Sherlock Holmes: Crimes & Punishments, from 2014. My personal favourite, as it introduces a brand new 'Deduction System', definitely inspired by the BBC's Sherlock.

All of the above are available on PC. On May 27, 2016, a new game will some out: Sherlock Holmes: The Devil's Daughter.

DID SOMEBODY SAY GAMING RECOMMENDATIONS OH LOOK YOU HAVE A PC WELL LET ME TELL YOU ABOUT WORLD OF WARCRA—

"Also, no more RPGs, please."

*pouuts and slinks off to the corner to sulk*

I guess that means recommending the ultimate time sinkMinecraft is out of the question, too, huh?