I'm playing Borderlands 2, but I have not had any legendary items drop. I have been told to buy the DLC, but I can not. Where can I farm for a legendary item, without having purchased DLC?
For easy loot, if you haven't already done it, I'd start by redeeming every single golden key I could and open a bazillion golden chests. Though, as mentioned in a comment, Golden chests only drop epic (purple) loot, not legendaries.
You can look here for a list of a ton of golden key codes (bottom of page) http://orcz.com/Borderlands_2:_Golden_Key (red codes are usually expired, black ones usually works, but not always, try sorting them by date and starting from the most recent)
Otherwise for farming, it depends on what you want, I'd suggest going through a list of legendaries (http://ca.ign.com/wikis/borderlands-2/Legendary_Weapons here for example, there are many online), finding one you like and farming it. Of course if you do not have any DLC, make sure the enemy that drops it is available in the vanilla game.
Here is a list of weaps and their location. Or you can go to Wildlife Preserve and farm loot midgets there.