In each installment of the Harry Potter series, Hermione Granger is the smartest of the trio. She has the best academic memory, highest grades, and always expanding her knowledge. However, it is Hermione's logic and by-the-book mentality that allows Harry to surpass her.
Harry is more likely to trust his gut instead of a textbook, and while he is not always right, it enables him to have a leg up in certain areas. It is Harry's instinct that manages to save his life on various occasions, something that Hermione has occasionally found herself lacking. These are ten times Harry's wit surpassed Hermione's cleverness.
PATRONUS
Harry Potter and the Prisoner of Azkaban is the first time the Patronus Charm is cast in the series. Remus Lupin uses it to send the Dementors away on their quest to find Sirius Black.
After multiple encounters with Dementors, and fainting after each time, Harry asks Remus to help him find a way to beat them. By the end of the novel, at thirteen years old, Harry can cast a corporeal Patronus Charm, sending away over a hundred Dementors. In his fifth year, Harry teaches the charm to Dumbledore's Army.
FLYING
Hermione is brilliant in nearly every class, Flying being a glaring exception. Only taken in first year, Hermione was not even able to successfully call for the broom to reach her hand. Meanwhile, Harry is a natural, the broom responding to him immediately, and without any previous training, he can successfully fly to retrieve Neville's remembrall.
Harry goes on to play Quidditch for five years of school, and it also how he survives the Hungarian Horntail during the first task of the Triwizard Tournament. Meanwhile, Hermione never took the same interest in flying, happy to keep her feet planted firmly on the ground. Hermione's fear would especially be amusing given it is her idea to escape Gringotts on the back of a flying dragon.
DIDN'T TRUST LOCKHART
Hermione's logic is rarely replaced by pure emotion. In Harry Potter and the Chamber of Secrets, Hermione falls for Gilderoy Lockhart's charm and books rather than the person he is. Harry does not fall for the trap and instead sees Lockhart as an incompetent professor. Lockhart's classes consist of him messing up spells and asking the students to take tests about himself.
Later, when Ginny is taken down to the Chamber, Lockhart plans to run away from the castle. When Harry and Ron catch him in the act, Lockhart admits he is a fraud and that he accomplished nothing he did in his books. Instead, he had obliviated everyone who had performed the deeds. Karma comes back to haunt him when Lockhart attempts to obliviate Harry and Ron, and only hits himself with the spell.
CONSIDERED THE DEATHLY HALLOWS
After seeing a drawing in the books Dumbledore left for Hermione and Harry, recognizing it as the symbol Xenophilius Lovegood had worn around his neck at Bill and Fleur's wedding, Harry, Ron, and Hermione go to investigate. Xenophilius tells them the story of the Deathly Hallows and the Peverell brothers, the original owners of the Elder Wand, Invisibility Cloak, and Resurrection Stone.
After hearing the tale, Harry becomes obsessed with the Deathly Hallows since becoming the holder of all three makes one the master of death. Hermione is against this, desperately wanting Harry to focus primarily on the Horcruxes. However, it is this consideration of the Hallows that makes Harry understand how important both are as well as the importance behind them when Voldemort is in search of the Elder Wand.
NARROWED DOWN LOCATION OF RAVENCLAW'S DIADEM
In Harry Potter and the Chamber of Secrets, a young, preserved, Tom Riddle speaks to Harry of all the ways they are similar. As much as Harry denied it and wished it not to be the case, in circumstances such as this one, their same thinking worked to Harry's advantage. With a war going on and running out of time, Harry is frantic to find the last Horcruxes.
As he tries to think of places they could be, he is struck by the answer, the Room of Requirement. Harry knows Voldemort, possibly all too well, and picks up his arrogance as a winning device. Tom Riddle would have placed a piece of his soul somewhere he would never think anyone would find it, the same place Harry had hidden his Potions textbook the year before.
TRUSTED HIS INSTINCTS ABOUT A HORCRUX IN HOGWARTS
Harry's connection to Voldemort gives him information he could not know otherwise. Even after his occlumency lessons, he is still unable to keep Voldemort out of his head. However, in Harry Potter and the Deathly Hallows, it works to Harry's aid.
Harry can feel Voldemort's emotions and get a look into what and where the other Horcruxes might be. When Hogwarts appears, Harry is adamant about a Horcrux being at Hogwarts while Hermione is not as sure. While they let the matter drop for a while, focusing on other possibilities, Harry is proven correct by the end.
LOYALTY OF THE ELDER WAND
Lord Voldemort had a theory about how the Elder Want worked. From the legend of the Deathly Hallows, the Elder Wand allows its owner to be unbeatable, and wizards would be willing to kill to get their hands on it. Voldemort believes that the Elder Wand changes loyalties based on whichever wizard killed its previous owner.
Under this theory, Severus Snape would become the wand's valid owner when he murdered Albus Dumbledore. Harry learns before Hermione that this is not the case. Instead, the Elder Wand changes loyalties based on whoever disarmed the previous owner. Meaning that instead of Snape, the Elder Wand belonged to Draco Malfoy until Harry disarmed him at Malfoy Manor. By the time Harry and Voldemort had their final battle, the Elder Wand belonged to Harry.
FAKED POTION FOR RON
Ron is not the most confident person, even in his area of expertise. When he is nervous about his upcoming Quidditch game, Harry pulls out his vile of Felix Felicis and pours it in Ron's drink. Hermione reprimands Harry and warns Ron not to drink it as they would both get in trouble.
Ron drinks it anyway, getting a confidence boost, and plays the game, helping Gryffindor win the match. As the rest of Gryffindor celebrates, Hermione tells Harry that giving Ron the potion was a bad idea, which is when Harry reveals the full vial of Felix Felicis. Hermione is shocked that Harry had managed to trick her into believing that he had given Ron the potion.
DRACO'S ALLEGIANCE
In Harry Potter and the Half-Blood Prince, Harry is convinced that Draco has joined the Death Eaters. However, he has no definite proof as Draco's left arm is always covered. Harry has two things to go on, having witnessed Draco in Borgin and Burkes and the theory that Draco is taking after his father, Lucius.
With no evidence, Hermione and Ron do not believe him, and Harry takes it amongst himself to follow Draco under the invisibility cloak. By the end of the novel, Harry is proven right. Draco does have the mark and had been working all year on a vanishing cabinet to allow the Death Eaters into Hogwarts undetected.
FINAL BATTLE
The culmination of everything Harry had been working for finally arrived during the Battle of Hogwarts. After Voldemort's request that Harry meets him in the forest to finish the war, one on one, Harry goes only after finding out one major piece of information; he is a Horcrux. Knowing that he must die for Voldemort to be defeated, Harry informs his friends to go after Nagini and walks to his death, ready.
However, his acceptance of what is in store for him is part of what protects his life and everyone else. After fulfilling the ultimate sacrifice, Harry tricks everyone at the battle that he is dead. Finally, when the time is right, Harry reveals himself to be alive and defeats Voldemort.