SPOILERS AS FOLLOWS
The only one I can think of is the ArchMage Hall in the College of Winterhold.
It is free (you must complete a quest line to get it though), but it is not really a house, more of a huge room, many chests, an ingredient garden (alchemy) a bed, and lots of loot and potions.
It also has an alchemy table and an enchanting table (plus lots of soul gems).
It has the same setting of a house, and is even bigger/better than most.
Also, it does not require any DLC to obtain it.
gaming.stackexchange.com/users/9608/Shadow Z.
You can get a free house in Solstheim if you have Dragonborn installed and you stop an assassination.
You could squat in the Abandoned_House or one of the other safe places .
gaming.stackexchange.com/users/1327/Zoredache
There is a glitch that allow you to get a house in town for free when becoming Thane: Talk to the steward about the house, say yes and then quickly go to the nearest container and place all of your gold before the steward finishes the dialogue.
You should get the book and key, now take the gold back from the container.
NOTE:
In all the city palaces, make sure to find the nearest container and use Slow Time shout if you need help.
Here are some near containers in Cities Markarth:
Dwarven chest next to the steward.
Solitude: Cupboard next to Falk.
Whiterun:
Drawers in the courtroom [Use Slow Time or Whirlwind Sprint], chest in the maproom, and other containers in steward's bedroom.
Windhelm: Containers in kitchen
[Use Slow Time or Whirlwind Sprint].
Riften:
Chests left and right of the court.
gaming.stackexchange.com/users/125410/Jim
By the way, you can wait and collect 5000 septims.
Talk to the steward to buy a home, and after confirming, quickly exit while he is still speaking and store all your money in the closest chest, barrel or cupboard.
When the steward has finished to speak, the game will try take you 5000, but he can't because you don't have them.
You can still get the key and enter the house.
gaming.stackexchange.com/users/64959/Vinz243
There are some free houses in the game, but you need to complete certain guild questline to get a free house.
College of Winterhold:
Arch Mage's Quarters.
Dark Brotherhood: Dawnstar Sanctuary.
gaming.stackexchange.com/users/127996/Jim Jones
This is similar to, but not the same as, the Arch Mage's Quarters mentioned in a couple answers already.
For that you have to go through an entire quest line.
If you merely start the College quest
line(gain access to the college)
, you get a cubicle in the Hall of Attainment with a bed and a number of safe storage containers excluding the barrels.
Across the way in the Hall of Attainment you can find both an alchemy lab with lots of free ingredients and an arcane enchanter with a number of free soul gems both empty and full.
gaming.stackexchange.com/users/181987/Jtenorj3
I just befriended Uthgerd the Unbroken
and she became my companion and gave me her house key.
Even if she dies you get to keep the house and store all your stuff in.
It's in Whiterun.
gaming.stackexchange.com/users/241482/Severus40k
I love to play Skyrim, but I am broke. I can't seem to get enough money for the cheapest one I know of, the one In Whiterun. Does anyone know of a cheaper one or a way I can get a free one?
Is there a way to get a free house in skyrim?
gaming.stackexchange.com
<the-elder-scrolls-5-skyrim>
gaming.stackexchange.com/questions/148104
gaming.stackexchange.com/users/65064/Zeke Macintyre
 There are some free houses in the game, but you need to complete certain guild questline to get a free house. The only one I can think of is the ArchMage Hall in the College of Winterhold. It seems that there are ways to get a free house in Skyrim, mostly as a reward after completing certain quest lines.
The community details few ways to obtain a free house. However a user is suggesting a glitch.