What fee does Sherlock Holmes ask for from the King of Bohemia? Why do you think he asks for that?

At the end of "A Scandal in Bohemia," the king offers Sherlock Holmes an "emerald snake ring" as payment for his services. Sherlock rejects this generous offer, however, and asks instead for the photograph of Irene Adler, something which he finds far more valuable.
In the final paragraph of the story, Watson gives us a clue as to why Sherlock chose the photograph over the ring:

"And when he speaks of Irene Adler, or when he refers to her photograph, it is always under the honorable title of the woman."

For Sherlock, then, Irene Adler is a woman who surpasses all other women. She is the only person who has (thus far) outwitted Sherlock, a feat she achieved using the medium of disguise, one of Sherlock's most famous methods. Before meeting Irene, Sherlock used to joke about women's lack of ability ("make merry over the cleverness of women") but Irene has forced him to dramatically re-evaluate his opinion. It is for this reason that he values her above any other, as Watson foreshadowed in the opening lines of the story:

"In his eyes she eclipses and predominates the whole of her sex."