The sentence means 'that boy' is making her (very) angry. You could write it as such; ~He is the most infuriating boy I have ever seen!~

Thanks for posting that nice long passage. Keep doing that when you post questions, it really helps. In this case I knew EXACTLY what it was the instant I saw the title. It's from "The Adventures of Tom Sawyer," by Mark Twain. Be aware that it was written in 1876, and Mark Twain is writing about his childhood in the 1840s. Mark Twain was a pioneer in trying to capture the way people really spoke. Particularly within quoted dialogue, you are reading slang, colloquialism, and "dialect" from the 1840s. Most NATIVE speakers just guess at a lot of the meanings from context. I'm guessing, but I'm pretty sure, that it "beat" means "surpass," "exceed;" in this case, surpass Tom in being generally difficult, disobedient, and mischievous. I've never heard "beat" used this way in real life, although there might be regions in the U.S. where it still is. It might have been regional even in 1840. If you go on to "The Adventures of Huckleberry Finn," you will find that in that book Mark Twain makes a point of saying that "In this book a number of dialects are used, to wit: the Missouri negro dialect; the extremest form of the backwoods Southwestern dialect; the ordinary 'Pike County' dialect; and four modified varieties of this last."