A good example of a functional character, Mr. Riley, makes his entrance early on in The Mill on the Floss. Riley comes to visit Mr. Tulliver about dam arbitration in book 1, chapter 3. While he is there, Mr. Tulliver consults with him about Tom's education. Riley recommends sending Tom to Stelling's school. This turns out to be disastrous advice, as not only is Tom miserable there, but he also doesn't learn anything that will be of use to him later on when the family has a financial crisis and he has to seek work. Although this is an important part of the novel, Riley himself doesn't play a significant role in the story beyond his discussion with Mr. Tulliver.
The most obvious choice of a character whose existence is an end in itself is Maggie Tulliver, as she is the main protagonist of The Mill on the Floss. However, other characters also fit this description, such as Mrs. Tulliver. At the beginning of the story, she is presented as a dim-witted, shallow woman whose negative opinions about her daughter reflect society's prejudice against strong-willed, intelligent girls. But when Maggie returns home in disgrace and Tom forbids her from entering the house in book 7, chapter 1, Mrs. Tulliver says, "My child! I'll go with you. You've got a mother." Her unexpected sympathy for her daughter shows she is more than just a one-dimensional figure who exists for only one purpose.