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?
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.
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.
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.
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.