When Pip finds out a secret benefactor is financing him to become a gentleman—the "great expectations" of the novel's title—his expectation is that this is the work of Miss Havisham. He expects that she wants to raise him up in wealth and social class so that he can marry Estella.

This expectation is utterly dashed: Pip learns that Magwitch, the convict, is his benefactor, and that Miss Havisham has nothing to do with his elevation. She has raised Estella to break hearts, not gain Pip's heart.

Pip also has the secondary expectation that his life will be better once he becomes a rich gentleman, but that expectation too turns out to be false. Pip only regains a sense of self and wholeness after he is humbled, loses his money, has Joe pay his debts, and comes to realize that living an honorable and simple life is better than being a rich snob.

None of Pip's expectations come to pass, but he nevertheless becomes a better, kinder, and wiser person due to his experiences.

I would say that Pip gets some of his great expectation, but not all of it.

The main part of Pip's great expectation is that he will become a gentleman and a man with money.  This part of his expectation comes true as he becomes much more (economically and socially) than he was at the start of the book.

However, Pip does not get this in the way he expects.  He gets it because of the help of a convict, not Miss Havisham.

In addition, Pip does not get Estella the way he thinks he will at the beginning of the book.  He starts out thinking they will marry, but that does not happen.

Pip, an orphan and adopted son of his much older sister and her husband, Joe, who is a blacksmith, has every reason to expect to live a humble, ordinary life as the novel opens. He himself may expect to become a blacksmith like Joe.

In this way, he is like the majority of young people in the United States today. He doesn't have $250,000 in the bank to pay for college, and he doesn't get large monthly checks from a trust fund. He has to expect to scramble into whatever life he can find.

But suddenly that all changes. An anonymous, wealthy benefactor is going to bankroll him. Suddenly, he has the chance to get the Victorian equivalent of a top-notch education, and a big monthly check is put in his pocket. Now he has "great expectations": with the benefactor and the money, Pip can expect to go far in life, to be important, not just an obscure laborer.

Pip assumes his benefactor is Miss Havisham, whom he thinks is grooming him to marry Estella. He couldn't be more wrong. His benefactor is the convict Magwitch, whom Pip helped at the beginning of the novel. He couldn't more horrified—but as part of his maturation, learns to come to terms with it and value Magwitch's good heart.

Pip expects himself to become a gentleman.  In doing so, he hopes to be wealthy, scholarly, mannerly, and deserving of Estella's love and admiration.

In another sense, others place great expectations upon Pip.  Magwitch sponsors Pip's education in London because he expects Pip to learn to be a better gentleman than Compeyson who was born a gentleman.  Likewise, Joe has great expectations for Pip because he tries to raise him to be hard working and kind. Biddy, at first, expects Pip to be able to do great things with his intelligence, until she sees him begin influenced by Miss Havisham and Estella.

Other characters in the book have expectations for Pip, but not necessarily "great" or "good" ones for the innocent protagonist.