When Esther came before the throne, the king promised to grant her request "even up to half" of what?

Answer: The kingdom

Xerxes asked, "What is your request? Even up to half the kingdom, it will be given to you," and he repeated the promise at her banquets.

““What is it, Queen Esther?” the king inquired. “What is your request? Even up to half the kingdom, it will be given to you.””— Esther 5:3 (BSB)