If you are on The Girl Who Played with Fire, you see the button labeled "Read first chapter FREE" in the green box (see below).
Example of a Kindle App Embedded:
But if you go to the Kindle edition for the same book, The Girl Who Played with Fire, there is no such box. Which seems a bit odd. The other options for sending the book to any of the supported Kindle Apps is there, but not the option of reading the book on the Kindle Web app.
One of the neatest features of the web app is that you can embed the Kindle Web reader on any web page.
|Embed on a web page|
|Email, or share via Facebook or Twitter|
|You can change the size of the text, line spacing, and even a color.|
Amazon.com Help: Kindle for the Web Beta
Free Reading Apps