Source: http://en.wikipedia.org/wiki/Predestination_(film)

If a time machine is created and somebody goes to the past, anything he changes in the past will affect the present time, and the person that travel will be changed by the butterfly effect which will make him cease to exist or forget the mission. The only person that will not be affected by changing the past is a person that doesn’t exist and that was never born by parents from the past.

How to create a person that doesn’t exist?

Cycle 1:
Year 0:
-biologically engineer a person with both male and female genitals (using hermaphrodite person’s dna)
-Have a surrogate mother deliver the baby.
year 1 the baby is born and is sent to the past
year 18: you send some guy to falsely impregnate the girl with sperm created from her own stem cells
year 19: during the delivery, remove the baby and the female organs, bring out the male organ, steal the baby
year 19-25: she will learn to become a man
year 27:
– convince her to go back to the past to seek revenge
– Replace the old father with her and make her fell in-love with the younger female self

Cycle 2:
Year 1: Send her own baby to the past.
Year 40: Make her/him do all the tasks that previously were done by existing humans in order to remove the born human interactions from the loop

Cycle3
Year 50+:
– you have an agent that is not affected by temporal causation
– use him to destroy targets by giving him fake future newspapers
– make his younger version kill him in order to close the loop

Repeat cycles to create a new agent for another period of time and city

This is a butterfly effect causality hack as long as you are careful enough not to disturb the history of the time machine inventor by making sure the timeline period is short and the location of the events are far.