Quality Fangames and Indie Games - Do you know any?

Indie:

Shank 2 - Damn it's fun, though it's super challenging

Super Meat Boy - Hard, fun

Fangamez

Megaman 2

Dwarf Fortress, Dominions.

Indie:

Retro City Rampage- Also known as "Grand Theftendo" was also released for PS3, Xbox 360, Wii, PSVita, and 3DS.

Bleed- A very difficult Platformer/Shooter game in a retro 8 bit style.

Knights of Pen and Paper +1 Edition- turn based RPG that also involves elements of Dungeons & Dragons. Also Released on Android.

Mutant Mudds: Deluxe Edition- Platformer in a retro 8 bit style. Also Released on 3DS.

The Deponia Series- Point & Click Adventure games, Edna & Harvey games also made by the same developer.

Giana Sisters: Twisted Dreams- Platformer

Guacamelee: Gold Edition - Platformer, also made for Playstation.

A repost from a different yet similar thread :

"Hero Core" for true retro styled metroidvania-ish shootings.
"Gamma Brothers" just because of awesome arcade pixelcore shooterness.
"Deep Within A Forest" for oddly relaxing exploration-platforming.
"An untitled story" for surprisingly epic exploration-platforming (give it chance, it grows in scale).
"Treasure Adventure Story" for similar exploration-platforming, albeit not my favourite of the last three.
"Toribash" for freeform fighting with a steep initial learning curve.
"Dwarf Fortress" if you have patience, enjoy truer roguelikes (ASCII/tile form with complex mechanics and controls) and have a sadistic streak.
"IVAN" for the Dark Souls equivalent of roguelike games, plus crazy humour. If you liked "Yume Nikki" and tried "Ib" as others have suggested, then perhaps look into "Mad Father" too. Not as good as Ib but still~
"Maldita Castilla" is another good classic-styled romp.

A really good fangame I've played is the Library Defense Game from Touhou? It's all Doujin I believe so it's free somewhere?

Ooh, neat list. I'm gonna have to check back here tomorrow and go through it. :D

Here's a bunch more from older comments and my searches.

Indie

• Toribash - Incremental fighting game.
• Yume Nikki - Earthbound on LSD.
• Ib -
• Dwarf Fortress -
• IVAN -
• Maldita Castilla -
• Off - Insanely creepy RPG.
• The Underside -
• Desktop Dungeon - Original version, Dungeon crawl Minesweeper.
• Ninja Senki - Ninja platformer.
• Rom Check Fail - Random gameplay mashup.
• Tower of Heaven -
• A Nation of Wind -
• Verge -
• I Wanna Be The Guy - Pretty sure you've heard of this. If not, it's the hardest platformer.
• Wilfred the Hero -
• Push! - Picross with a story.
• Postmortem: One Must Die - Basic Version, narrative adventure where your actions change the ending.

Fangames

• PatchCon: Defend the Library and English Patch - Touhou Fangame
• Street Fighter X Mega Man - Megaman Streefighter crossover.
• Home - OFF Fangame spinoff

Three the Hard Way, by far the best game made with the RM2K (RPG Maker 2000) imo. Great dialogue, great story, simply worth playing. Nowadays it would be sold on Steam I'm sure - and it would be worth buying.

I Wanna Be The Fangame

One of my favorite indie games are Nitronic Rush, with its sequel; Distance, in the works and available in alpha through a pre-order.
I recommend trying Nitronic Rush and if you love it through the end then buy into the alpha of Distance.

You know, that OFF looks a lot like Undertale.

Undertale isn't complete yet, and will be a pay-for game when released, but there is a demo that is well worth playing just for the scenario it creates. Just don't hold me responsible for the pre-orders it will likely generate :3