If Pip has a middle name, it is not given in the novel. He introduces himself in the very first sentence of Chapter 1.
MY FATHER'S FAMILY name being Pirrip, and my Christian name Philip, my infant tongue could make of both names nothing longer or more explicit than Pip. So, I called myself Pip, and came to be called Pip.
So Pip's name is Philip Pirrip, but he calls himself Pip and is called Pip by everybody, including Miss Havisham when he eventually meets her. He becomes stuck with the name when Mr. Jaggers informs him that he is to be educated as a gentleman by an anonymous patron who stipulates that he must always retain the name of Pip. Therefore he becomes Mr. Pip. His first name, or nickname, becomes his last name. He believes this stipulation is only a sign of Miss Havisham's fondness for him as well as an indication of her eccentricity. The real reason is that Abel Magwitch wants to be able to find him if and when he is able to return to England. There should only be one person named Mr. Pip in all of England, so Pip should be easy for Magwitch to find. When Pip first encounters the terrible convict in Chapter 1, he is forced to repeat his name to the man three times, which will ensure that Magwitch will remember the name and remember the boy by that name.
“Tell us your name!” said the man. “Quick!”
“Once more,” said the man, staring at me. “Give it mouth!”
“Pip. Pip, sir.”
Phillip Pirrip. he named himself pip because when he was a baby he couldn't pronounce his name so he gave himself a nickname Pip.
Pip's full name is Phillip Pirrip, but when he was a child he couldn't pronounce Phillip so he called himself Pip and it stuck.