Borges has a short story about a labyrinth of words; a mammoth novel with the chapters scrambled in order. It's your job as reader to piece together the narrative, and take whatever you will from it. But since the narrative itself can go in multiple paths, different readers will read it very differently.
The idea of a novella or novel which defied order has really intrigued me. In its simplest case, imagine a novel that you could read either forwards or backwards. I would love for this to be a murder mystery. In one direction, it is the story of the detective figuring out what happened. You start with an unsolvable case, and talk to people and go bit by bit until you know what happened. In the other, it is the story of the criminal erasing clues and evading detection. You start with the crime, and visit each scene to ensure your safety, ending with an unsolvable case. I don't know if it's doable, but it's a cool idea nonetheless.
Things are certainly interesting when you want two paths through a novel, but even so you can determine an interior order suitable for either. Making a novel in which any chapter can be read in any order, without having it devolve into a series of short stories, that seems even more difficult. A novel, by definition, it seems, has to have a unifying thread that runs through all of it. It isn't mix-and-match. How can you have character development, large-scale themes, climax and resolution if you need to be able to take any chapter out of its place and put it anywhere else?
My first idea was to write a novel like an essay, or at least vaguely. My idea was that in most of my essays for class, slight alterations in the first and last paragraphs mean that each interior, supporting paragraph, can be displaced and rearranged at will. Of course, I generally think there is a good way and a few bad ways to rearrange the paragraphs, but the point of the essay would still come across no matter the order. Extrapolating this, if I work from a first and last chapter idea, I can introduce an idea in the first/last chapters, and have each other chapter pull and tug at the idea, flesh it out in interesting ways. They have a common thread, which is the idea, but each chapter is a unit unto itself and can be rearranged at will. The idea could be anything. It could be a murder mystery again, with the first and last chapters being the detective saying what happened and/or the crime itself, and each intermediate chapter being one piece of evidence that leads on the path from one to the other. It could be simpler, the description of a world or the different things that happen in a specific building.
But that feels too much like a short story anthology, not enough like a real novel.
Another idea I had was based on Italo Calvino's "If on a winter's night a traveller" (I think), a novel which is in effect the start of 14 or so different novels. In that structure, order doesn't matter because the novel is about the experience of reading, of writing, and of creation -- not about the plot of any of the 14 or so different novels begun in the book. However, even here, Calvino has a narrative plot with two readers that strings all of the works together, to keep it from being too much of a short-story anthology. But the idea is there: If the novel is about the creative experience, writing and reading and so forth, then the thread can exist between stories and not necessarily need a linear plot?
Which brings me back to something I have always enjoyed (and those of you who read my earliest writings will attest to this), which is the difference between fantasy and reality, and how stories mediate this gap. The first long story that I remember writing (I've been writing since I could read), was full of that surreal stuff. First we (my cousin and I) wrote a story, and then we wrote a story about the story, and then we wrote the combination and resolution of the two. And then we realized that we still had a lot of loose ends, so we tied all of them up. The point of it (for me) was how imagination works in the world, how the creative process works, how characters become real to an author and how the story threatens to swallow you whole - for better or for worse. It was a really cool story. What I liked best about it was the recusive aspect, the self-referential aspect, and it meant that you could, in principle, read the first two stories in either order and it would make just as much sense either way.
So it seems to me that writing about the experience of reading and writing is probably the best way to do away with the linear quality of stories. The scheme works this way:
1) Simple story, within the paradigm: brilliant inspector solves murder mystery
2) Meta-story: brilliant murder mystery writer solves murder mystery
3) Alternate Meta-story: clever murder mystery writer commits murder, is caught OR murderer is caught by murder mystery writer
4) Self-referential Meta-story: murder mystery writer sees murder mystery going on around him and writes it
5) Surreal self-referential Meta-story: murder mystery writer writes murder mystery and sees it going on around him
6) Alternate surreal self-referential Meta-story: Criminal writes murder mystey which leads to his own arrest
etcetera etcetera etcetera. You could use any plot/paradigm. The structure, the iteration, and so forth, draw attention away from the similarities (the paradigm itself) and point out how each story is different, i.e. how the story itself effects each iteration. You can even keep names the same - or very similar - between iterations (i.e. Gene, Jem, Jim, James, John, Jon, Jonas, Jonah, Jonathan, etc.)
So. Yeah. Recursion makes stories better, and seems like the best way to make an unordered story (at least to me).