WARNING: Game spoiler details follow
Despite having no real plot or structure, Yume Nikki does in fact have an ending scene.
The existence of the ending is only alluded to in-game, with the cryptic instruction to 'drop effects in the door room', although there is no indication of what the purpose of this is.
At first, the simple act of dropping all the effect in the Nexus has no effect, and there is no indication that anything has changed. Only when the player ventures out onto the balcony in the real world does the ending become apparent.
The Ending sequence
Collect all 24 effects in the game, then return to the Nexus. Drop all of your effects, then pinch yourself awake. After Madotsuki wakes back up, go out to the balcony. A set of stairs should have appeared. Walk up them, and you will see Madotsuki will plummet to her death, thus ending the game.
The ending scene is a black screen with a bloodstain in the center, followed by two of the jellyfish creatures walking in and standing there for the remainder of the credits. The Jellyfish can be interacted with on this screen. They can be made to chime with the number keys. The lower Jellyfish reacts to the (1) key and makes a low sound, and the upper Jellyfish reacts to the (3) key and makes a higher-pitched sound.