Despite having no real plot or structure, Yume Nikki does in fact have an ending scene.
The mere existence of the ending is only alluded to in-game, with the cryptic instructions 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 effects in the Nexus appears to have no result, and there is no indication that anything has changed. Only when the player ventures out onto the balcony in the real world does the nature of the ending become apparent.
The Ending Sequence
After collecting all 24 Effects in the game, Madotsuki must return to the Nexus. She then drops every last one of her effects (by equipping them and pressing the '5' key).
Madotsuki then leaves the Dream World by pinching herself awake. She explores her room, and finds that a set of stairs have appeared on the balcony, leading over the railing.
Madotsuki climbs the stairs, jumps off the side of the balcony and plummets downward, then everything becomes dark.
A heavy, echoing thud breaks the silence, and a pool of blood fades into view. Madotsuki is dead. Two Jellyfish from her dream world approach from different directions and surrond her dead body as the credits appear, and the mournful ending theme begins to play.
Trivia
The Jellyfish can be interacted with on the ending screen, pressing the number keys causes them to chime.
The lower Jellyfish reacts to the (1) key and makes a higher-pitched sound, and the upper Jellyfish reacts to the (3) key and makes a lower sound.