Final answer:
The Mariner's offense in The Rime of the Ancient Mariner is the killing of an albatross, an act that brings misfortune upon him and his crew.
Step-by-step explanation:
In The Rime of the Ancient Mariner, the great offense the Mariner commits is that he kills an albatross. The albatross is considered to be a good omen by his fellow sailors, and its unnecessary killing brings about dire consequences for the Mariner and his crew. This act of thoughtless violence against an innocent creature symbolizes a deeply moral and spiritual transgression within the narrative of the poem.