Last year, my son and I went down to St. Louis for a Cub/Cardinal game. I happen to know a fellow official who lives in the St. Louis area, and this guy also pitches batting practice to some of the Cardinals before every home game. We decide to show up early enough for BP. So my son, who is 11, and has only been to 4 or 5 games in his life, and I are standing behind the St. Louis dugout watching BP, when the official I know finishes his part, and comes over to the dugout. He sees us, and gives my son a BP ball and one of the little Gatorade bottles the players use. My son was so excited about the ball, he kept pulling it out of his pocket and looking at it all game. He brought it home and put it one of those ball cases and keeps it in his room. (It is the only Cardinal-related item allowed in the house...)
Me, I've been to many, many games in my lifetime and have never, ever gotten a ball from a game. Not that I'm bitter or anything.