So you want to download everything from your Facebook account? This is the unofficial method.
This is what you should see, with everything you've ever posted and shared on Facebook below. Notice the "All" drop-down menu in the top right.
Click it and select the type of content you want to download from your Facebook account. If you want to download everything (this will take a very long time), leave the selection on All. For the purposes of this guide, I'm choosing Info Updates because it doesn't contain a lot of data.
Scroll to the bottom. I don't mean choose the month you joined Facebook. I mean literally scroll all the way down, making sure that everything loads in your browser.
Once everything is loaded, you'll want to save it with your browser. In Chrome, click the wrench icon and choose "Save page as..." In other browsers, look for a similar option in the "File" menu or equivalent.
Name the file anything you'd like. Make sure the "Web Page, Complete" type is chosen, or equivalent depending on your browser.
Navigate to where you saved the webpage. You should see something like this. Click on the HTML file you've saved to open it up.
You should be back at your Activity Log. Notice, however, that this is a local file: you don't have to be on Facebook to browse everythign as it's now stored on your computer. The links will still take you to Facebook, which requires being online, but everything else is now stored offline.