The goal of the Greek hero was kleos (glory), which literally referred to other people speaking well about you. Kleos could especially be obtained by killing and finally being killed in an honorable fight. The ultimate form of kleos was called kleos aphthiton (‘imperishable glory’), which resulted in being remembered for all time. This was the closert the mortal Greek could come to immortality.

the great world history book stephan dinkgreve ancient greece homer iliad odyssey imperishable glory kleos

"The Great World History Book"