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.
Waht she said ^^