What is the State of The Art of Writer AIs (Deep Learning)?
Researchers are trying it out, but RNNs learn character-by-character (sequences of characters), so it is difficult to get something that resembles a story plot, as a whole. This link by a Stanford researcher explains the current (2015) state of the art. Here is the code and here are some data sets for you to start experimenting with.

Another experiment involving the writing of a travel blog also concluded that you cannot really create a long passage that makes sense at the moment. The recommendation was to look at the word level, rather than character, and focus on something more manageable, such as sentence autocompletion.

When it comes to words, the more unique ones there are in the source data set (for example using the Game of Thrones books), the tougher it becomes to train a good model. Suggestions are to limit input to more basic words (think children's vocabulary) and to have at least a total training sample at least 100 times larger than the desired output.

A trained neural network could perhaps output smaller texts (&lt;10000 words) that make some sense, if they are of a rather structured nature. A whole book needs both a coherent plot from start to finish, as well as twists in-between. As a result, it is still way too difficult.

This guy tried it out and discovered that the AI would get sometimes stuck in loops. Furthermore, the draft produced did not always make sense in terms of the storyline and required heavy human editing.

Here is another example of an LSTM RNN outputting a Harry Potter chapter, where the sentences are grammatically correct, but sometimes make no sense.

If you want to learn more about RNNs, the Deep Learning book by Goodfellow, Bengio, and Courville comes highly recommended and has a relevant chapter.

For something more specific to writing, The Bestseller Code book uses text mining techniques and should be an interesting read.