Some that I like:

一石二鳥 (issekinichou ; literally: one stone, two birds ; I'll let you guess the meaning (yes, it's very close to an expression in English))

自業自得 (jigoujitoku ; reap what you sow)

自作自演 (jisakujien ; self-made self-staged ; for example, used for the act of creating a staged situation and then acting as if it were genuine)

自問自答 (jimonjitou ; answering one's own question)

自暴自棄 (jiboujiki ; self-abandonment)

自画自賛 (jigajisan ; singing one's own praises)

自給自足 (jikyuujisoku ; self-sufficiency)

自由自在 (jiyuujizai ; free/unrestricted)

(Yeah, I have a thing for the 自◯自◯ ones)