Netflix app is free. Swapnote is also. There's a Kid Icarus video montage that may or may not still be available. Nintendo Video is a free app. That's really about it. You can check the demo section and get those for free but there isn't that much.
"I'm saying people outside GameFAQs didn't care when Mega Man Legends 3 was cancelled." horror_spooky 3DS FC: 1977-1514-4983, PM if you add me
Well, for sure there's Swapnote, Nintendo Video and Netflix. As well as game demos. You used to be able to get Zelda Four Swords, Pokedex 3D (not the Pro version), and 3d Classics Excitebike for free but those ones are not longer there.
On the eShop's main menu, scroll left to "Search." If you scroll down, you'll find the price filter, set to "Free - Any." Change "Any" to "0" and hit "Apply Filters." This will give you all 3DSWare that comes free of charge.
On the main menu of the eShop, you can usually also find a "Demos" category to the right. In there, you can find lots of free demos of various games.