That would be Leanpub. It has the express purpose to publish works in progress, and also provides a store front where readers can buy your book right away and receive free updates as you publish new chapters. Epub, mobi and PDF are all supported. You retain the copyright, and there's no exclusivity, so you can publish elsewhere when you're done. Last but not least, your royalties are about 90% of the book price.
I have a couple of books there at different stages (one with a co-author, which is also supported) and can only say good things of them and of their support forum. The only two caveats: starting a book will cost you 99$, and their main workflow uses a markup language called Markdown which is simple but you might not be familiar with. Importing your Word files is also supported, though.