OK, it's a macabre question, but I see what you're getting at.
Going back to amputation, the original heading of your post, given that blood loss could be stemmed by cautery etc, you could remove all four limbs one after the other. This has been done to victims of land mines etc in the past.
So then your left with the torso, head and neck. The trouble is, where do you want to go from here? Removal of large areas of skin, the body's largest organ, would lead to death in a few hours for various reasons, one of which is again shock.
I'm not quite sure what you mean by "basic parts of the body". Normally "basic" refers to something of a very low or simple level - rudimentary, in other words. It's not possible to regard body parts in this way as they have unique, exclusive functions often not dependant upon other "parts". Your concept of basic body parts may not be the same as another persons and consequently, the lack or presence of them means different things to different people.
If you're talking about internal organs, that's an entirely different ball game depending on the organ. Remove the liver or kidneys and you'd be dead within a couple of hours. Remove the thyroid gland and you'll survive adequately as long as you're given daily doses of L- Thyroxine.