Touhou
The Touhou Project is a Japanese doujin bullet-hell shooter game series entirely made by the one-man Team Shanghai Alice. The author, ZUN, made all the graphics, music, and programs by himself, and releases the games in a humble stand at Comiket. The games themselves are extremely challenging, yet highly addicting.
What puts this beyond any normal doujin work is it's gigantic worldwide fanbase, who elevate the game's basic characters into viral memes. Thousands of doujin works and communities have been built upon these memes. The result is an unparalleled cultural influence on the internet, akin to that of the Greco-Roman gods on Western Civilization.
Full Article: Gensokyo
The Touhou games have some marginal canon storylines. However, it's the world within Touhou that really shines.
Full Article: Touhou Characters
Touhou characters are have a very basic structure. All they have (in the series, anyway) is a basic, one-liner personality with a few quirks, inferred by their lines of speech. However, these easily memorable traits allow the characters to transcend any storyline and put themselves into the thoughts of the gamers. The personalities of the characters lend themselves to innumerable interpretations and few boundaries, allowing creative fans to build any story they want using these characters.
There's been a bunch of discussion on how and if Gensokyo could actually exist, albeit away from sight in some way. In fact, Well, if it did exist, this commenter has some copypasta on the theories behind it.
Look at it this way.
On most browsers, you can bring up your browsing history by pressing Control-H. (No, this is not going to become a discussion of werecows.) On Firefox, this brings up a sidebar that shows up on the left side of the window. If you put your mouse over the edge of the sidebar, the cursor will turn into a different kind of arrow. By clicking and dragging it, you can move the edge of the sidebar back and forth. You are, to put it another way, manipulating the border between the normal window and the history window. By moving the mouse, you can increase the portion of the window devoted to either part. In a more extreme view of this situation, you're increasing or decreasing the amount of existence the sidebar has. Now, let's apply this idea to something more abstract. Look out your window. If you don't live in a highly urbanized area, you should be able to see the horizon. Think of this as the border between the land and the sky. The land and sky are obviously distinguishable thanks to this boundary. Now, if you were to "drag" the sash between the sky and the land, or to manipulate the border between land and sky, you would end up causing the sky to become larger and the land to become smaller, or vice versa. An effect of this might be to cause something that was just on the ground to suddenly be hundreds of feet in the air. Truly a frightening situation to be in. So, look at it this way - manipulating the border between two physical things shifts whatever balance there is in the interaction between those things. Alternatively, by manipulating the border between two things, you can change the manner in which they exist.
Still, this isn't that abstract, since it's still dealing with real things in the real world. Many believe that in this world, there are those things that are true, and those that obviously aren't. This divides reality into two extremes: truth and falsehood. But, since we have two extremes, logically one can imagine a boundary between those two extremes - the border between truth and lies. If one were to manipulate this border, suddenly things that were pure fantasy (flying pigs, for the sake of argument) have become reality - or things from reality have ceased to exist. This is how Yukari is said to have invaded the moon - by manipulating the border between truth and lies, as applied to the reflection of the moon on a pond, she was able to make the reflection of the moon into a manifestation of the actual moon, and so send her youkai army onto it. This is what's truly amazing about Yukari's power - the ability to manipulate the border between completely abstract concepts allows her to fundamentally change reality as we know it (at least in terms of two abstract concepts).