In Ch.18, "in the fourth year of Pip's apprenticeship to Joe, on a Saturday night" Jaggers meets Joe and Pip at the inn of the 'Three Jolly Bargemen.' Jaggers does not recognise Pip, but Pip recognises Jaggers as the gentleman whom he had met at Miss Havisham's house on his second visit. Jaggers tells them that he has some important information to convey to them in private. So the three of them proceed to Joe's house.
As soon as Jaggers announces that "Pip has great expectations," Pip records his astonishment thus:
"Joe and I gasped, and looked at one another."
Once, Pip had got over his initial shock and surprise Jaggers tells him that his benefactor wants him to be better educated. Pip replies "I said I had always longed for it."
So we can safely conclude that Pip's greatest desire was to secure a good education which would in turn help him to become a 'gentleman' and enable him to marry Estella.