Obviously, Boo's greatest display of courage was when he came to Jem and Scout's rescue near the end of Harper Lee's novel To Kill a Mockingbird. He apparently had been keeping an eye on Jem and Scout at night--Miss Stephanie claimed to have seen him prowling after hours on at least one occasion--and Boo came armed when he met up with Bob Ewell. Boo stood by the Finches after Ewell's death, and he must have known that an investigation would follow. Despite his need for privacy, he was ready to answer to his good deed if necessary. Luckily, Sheriff Tate's decision to call Ewell's death accidental saved Boo of a public unveiling.
In "To Kill a Mockingbird" Boo has been shut away from society all of his life with the exception of when he was very young he would sometimes be seen with his mother. He begins to put things in the tree as a means of reaching out to Scout and Jem. It took courage for him to try and reach out to another person. After Scout tears his shirt and leaves it behind, Boo mends it. Again he has stepped out of his safety zone.
The evening the children are attacked by Mr. Ewell Boo was following. We are not sure why he was following, but to confront Mr. Ewell and save the children meant that Boo would have to have physical and direct contact with another human being besides his brother. Aside from the courage it took to thwart the attacker and protect the children, it took courage for Boo just to be in the presence of others. He does this again when he takes Jem to his father and when he stays in the house and later allows Scout to take him to Jem, sits in the room, and walks home holding Scout's arm.