The difference between 자기, 자신, & 자기 자신?

The following words can act as reflexive pronouns in Korean:

• 자기자기 생각만 한다.
  You think only of yourself.
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• 자신네 자신이 생각하라.
  Think for yourself.
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• 자기 자신(너) 자기 자신도 돌보지 못 한다.
  (You) cannot look after yourself.
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• 저그 놈은 저밖에 모른다.
  He thinks only of himself.
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• 당신대통령께선 당신을 돌보지 않고 나라 걱정만 하십니다.
  The president neglects himself and only worries about the country.
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• 스스로문이 스스로 열렸다.
  The door opened by itself.
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• 서로(그들이) 서로 욕하다.
  (They) call each other names.
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• 자체목적 자체는 훌륭했다.
  The motive itself was admirable.

The Differences Between the Different Pronouns
자신 and 자기 are both reflexive pronouns, but 자신 has two functions that 자기 does not. 자신 can be used redundantly to emphasis the preceding noun or pronoun, but 자기 cannot. See the following example:

• 나 자신은 이 그림이 좋다.
  I, myself, like this picture.

자신 can also be used to mean "alone" or "on one's own," but 자기 cannot. 스스로 can also be used in this way. See the following the examples:

• 네 자신이 생각하라.
  Think for yourself.
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• 그는 스스로 결정했다.
  He decided for himself.

저 and 당신 are first person and second person pronouns, respectively, but they can also be used as reflexive pronouns, which can be very confusing. When used as reflexive pronouns, they function the same as 자기 and 자신, but 저 seems to be less respectful, and 당신 is more. Consider these examples:

• 그 놈은 제가 하지 않았다고 주장한다.
  That guy claims he did not do it.
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• 대통령께선 당신을 돌보지 않고 나라 걱정만 하십니다.
  The president neglects himself and only worries about the country.

Notice how the reflexive pronoun, 저, just like the first person pronoun 저, changes form when the subject marker 가 is attached. By the way, in the past, 저 and 당신 were used only as reflexive pronouns, but now they seem to be used more as first and second person pronouns than as reflexive pronouns.
자기 자신 is treated as an independent reflexive pronoun, and can replace 자신 and 자기 in many situations, but in complex sentences, there is a big difference in meaning. Consider the following examples:

1. 갑수는 을수에게 병수가 자기를 때렸다고 말했다.
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2. 갑수는 을수에게 병수가 자기 자신를 때렸다고 말했다.

In Sentence 1, is 갑수 saying that 병수 hit him (갑수) or that 병수 hit himself? In the case of Sentence 1, 자기 usually refers to the subject of the main clause, but not always, which makes it somewhat confusing. However, 자기 자신 is used only to refer to the subject of the clause it is in. Here are the translations:

1. 갑수는 을수에게 병수가 자기를 때렸다고 말했다.
   Kap-su told Eul-su that Byeong-su hit him (Kap-su).
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2. 갑수는 을수에게 병수가 자기 자신를 때렸다고 말했다.
   Kap-su told Eul-su that Byeong-su hit himself.

자체 is used as a reflexive pronoun for things or concepts. Consider the following example:

• 덕을 행한다는 것, 그 자체가 보상이다.
  Virtue is its own reward.