Which leap of emotional growth was the most for Emma and why? How did the word games and misunderstandings affect this in Emma?

Emma matures as a result of the events surrounding the trip to Box Hill and the word games.  She begins to understand her place in the community and how privileged she is, learning from her mistake, Emma's awareness about herself, her feelings for Mr. Knightley and the less fortunate are changed.  She looks at her actions and how they affect others in a more serious way, not as merely a game to entertain herself.

Emma experiences a true sense of freedom when she journeys to Box Hill on a summer afternoon with the group which includes Frank churchill, Jane Fairfax, Mr Knightley and others.  She finally gets to travel, a long held wish and she is very happy and sure of herself on this trip.  However, during the course of the word games, she openly insults Miss Bates who also accompanies the party, mostly to respond to Frank's teasing and flattery.  By all accounts, Frank's behavior towards Emma suggests that he is in love with her, but she, I think is just enjoying the attention, especially in this group of people.

During the word games, Emma is careless of the feelings of Miss Bates, even though she and Churchill enjoy a laugh, Mr. Knightley is furious with Emma for her inconsiderate behavior towards Miss Bates. As illustrated in this passage from the book:

"Oh! very well," exclaimed Miss Bates, "then I need not be uneasy. 'Three things very dull indeed.' That will just do for me, you know. I shall be sure to say three dull things as soon as ever I open my mouth, shan't I?—(looking round with the most good-humoured dependence on every body's assent)—Do not you all think I shall?"

Emma could not resist.

"Ah! ma'am, but there may be a difficulty. Pardon me—but you will be limited as to number—only three at once."

Miss Bates, deceived by the mock ceremony of her manner, did not immediately catch her meaning; but, when it burst on her, it could not anger, though a slight blush shewed that it could pain her.

"Ah!—well—to be sure. Yes, I see what she means, (turning to Mr. Knightley,) and I will try to hold my tongue. I must make myself very disagreeable, or she would not have said such a thing to an old friend." (Austen)

By the end of the day, Emma has realized her mistake, but the cost is very high, Mr. Knightley is disappointed and very angry with her.  His expectations, his hopes for Emma seem lost.  She is distraught and experiences a surge of emotional growth, maturing in her understanding of the world around her, she lives in an insulated world, safe in her rich home, while Miss Bates, is not so fortunate.   She is especially made aware of how highly she regards Mr. Knightley's approval and without it how despondent she feels.  She is coming to a greater understanding of her feelings for Mr. Knightley.

She is depressed and unhappy after the trip to Box Hill and worried that she has lost Mr Knightley's good opinion and the good opinion of the community by her thoughtless behavior towards Miss Bates.  She is genuinely concerned about the consequences of her actions, something that she has not really thought about before, she is not so sure of herself after the Box Hill trip.  She is humbled.