While not "permanently free," you can legally watch the Extended Edition for $0 using a timed trial.

The film takes place in the Shire, where Bilbo Baggins (Martin Freeman), a hobbit, lives a comfortable and peaceful life. However, his life is disrupted when the wizard Gandalf (Ian McKellen) and a group of dwarves, led by Thorin Oakenshield (Richard Armitage), arrive at his home. The dwarves are on a quest to reclaim their treasure, which was stolen by the dragon Smaug. Gandalf convinces Bilbo to join the dwarves on their perilous journey to the Lonely Mountain.

For those who give up on the "free" search, it is worth noting that the Extended Edition is often available on physical Blu-ray for less than the price of a cinema ticket. Second-hand stores (Goodwill, CeX, or eBay) sell used copies for $3–$5. For the price of a coffee, you get the film plus never-before-seen appendices (behind-the-scenes documentaries that are 9+ hours long).

Searching for "The Hobbit An Unexpected Journey Extended free" usually leads to pirate sites. These sites are notorious for:

: A new scene showing a young Bilbo at a Market Fair meeting Gandalf for the first time.