Agnishri is the term used to refer to a person who has conducted Athirathram yagna in Hinduism. Akhithiri is a corrupted form of Agni Shri.
Agyadhanam, Agnistoma and Athirathram are the three yagnas. They are also known as Adhanam, Yagam and Agni.
The yagnas have to be completed in order starting from Agyadhanam followed by Agnishtomam and finally Athirathram.
People who perform the yagnas in order are known as Adithiri, Somayagi or Chomathiri, and Agnishri or Akhithiri.
Akhithir means a person who has successfully completed the three yajnas.