Pepperminty Wiki is a complete wiki engine contained in a single file, inspired by @am2064's Minty Wiki. Upload to a PHP host and that's it. No database required. Uses Markdown. Has full text search and edit history support. Even has comments. Has a rudimentary API right now.
GetSimple is a content management system that uses all flat files on the back-end - no database is required. It's fast to set up at pretty much any web hosting provider or on any web server. The user interface is clean and the undo function will let you revert nearly any change you make to the site. Even though the files are XML on the back end you interact with the site through a web browser, just like you'd update a site built using any other CMS. If you want to set up your first website this might be a good package to start with.
Personal bookmark service. Does not use a database. Can be run entirely privately. Has an API and a full-text search function. Very simple to use. Also lends itself to snippets, note taking, comments, hashtags. Offers RSS and ATOM feeds.