가져다 두다 = 가져다가 두다 = 가지어다가 두다 = 가지다 + -아/어다가 + 두다.
-아/어다(가) introduces the fist part of an action sequence which usually involves or preparing or readying something for the subsequent part.
가지다 means to have, take, or own something when used by itself, but in the -다가 form, it indicates a concrete action of taking, bringing an item to do something with it, or to it.
가지다가 is wrong because it doesn't include the required 아/어 in it. Some verb endings attach directly without 아/어 (-면, -고, -지만, -려) while others require it (-아/어서, -아/어도, -아/어다가). You can identify them by checking how 하다 conjugates with them: 하면, 하고, 하지만, 하려 vs 해서, 해도, 해다가 - 하다 is a special case where it takes 여, as in 하여도, which then contracts to 해도.
(There is another -다가 with a different meaning that doesn't require 아/어, but it doesn't work well with 가지다.)
가져다가 is usually shortened to 갖다가 in conversation.
- 강아지가 뼈를 물어다 놓았다 = The puppy scavenged a bone from somewhere.
- 그 돌은 갖다가 뭐하려고 해? = What are you taking/bringing that rock for?
- 좋은 물건을 갖다가 다 망쳐 놨다 = He went and made a mess of a nice thing.