If you are searching for the cheapest instant camera in market then Fujifilm Instax 210 is the cheapest one you can find. I'm not counting those old Polaroid cameras you find in ebay and yard sales for $20. Those cameras are only good as collectable items not for day today use. Because most those cameras have no instant film avaliable and also those three Polaroid models which has film available cost you around $3 a photo print.
Instax 210 cost you only around $60 at Amazon. I think it's the best price you can get anywhere both online and local camera stores. In almost all other online stores it cost you around $100. This camera uses Fujifilm Instax Wide film only. So make sure to buy the Instax Wide type film not Mini film. Cost per photo print is around $0.50 just like Mini film.
It's little bigger than Instax Mini series cameras so it's not for the hand bag. 210 uses four AA batteries for power. I highly recommend you to buy couple of sets of Sanyo Eneloop AA Rechargeable batteries along with the camera if you are going to buy this one from Amazon. Those batteries last very long and self discharge rate is low too. Means those batteries won't discharge soon when you keep those without using.
These instant photo cameras are mainly focus on printing photos rather than photo quality. It's same for this model too. It will take great outdoor photos when there are good lighting conditions. When you take indoor photos makes sure to use Flash always. If not photos might get blur. There are several buttons on side the camera which let you control brightness. Flash kicks in automatically in low light conditions.
You can see in Above photo that you will get a close-up lens adapter along with the camera. It's great for self portraits. It even has a small mirror to show you how the photo should look. Flash take around 30 minutes to recharge and Photos might take around 1 minute to fully develop once it comes out of the camera.
Another important thing is never open your film packs before you use those. Also make sure to check the expire date on film packet and store those in a cold place where it won't get too much heat. You don't need to keep these film packs in fridge like old Polaroid film packs.
Instax 210 camera might be too big for you if you are search for a small camera. But if you need bigger 3 x 4 format photos then this is the cheapest model available in market today. There is Polaroid z340 digital instant camera but it cost you around $250 at Amazon.
This video shows you how to operate the camera and insert Film pack into the camera,