I never know how to review books like this. On one hand, some chapters were brilliant and got my mind spinning in a way that most books don't. On the other, my god was most of this book a slog. It suffers from the usual pop-sci problems: dumbed-down enough to not really represent the subject very well, but not far enough to be interesting to a layperson. As such it wobbles uneasily in limbo; taking four of its thirteen chapters to describe different kinds of multiverses---most of which turn out to be indistinguishable in the end :/
The first two chapters are quite good, and answered some of my burning curiosities in life. "How do we /know/ the universe is 14 billion years old?" It also presented quantum lensing in a way that really connected with me, along with some tests we can run to convince ourselves about it (though, interestingly, not tests we can ever tell anyone else about. Quantum weirdness at its weirdest!!!!)
It ends in a philosophical mess of "maybe our universe IS REALLY MATH" and kind of waffles its way through what this means and what it would look like to humans. As best I could tell (I started skimming around here because boooooooring) the argument double- or triple- counts the anthropic principle as evidence for WE ARE REALLY AND TRULY MATH. And even if it does, like, who cares? There doesn't seem to be any testable hypothesis here, and Tegmark's beliefs don't seem to be paying rent.
I'll leave you with this: if you're REALLY EXCITED about this book, you'll probably find something of value here. Don't be afraid of skimming along though; Tegmark sure could have used an editor who was sympathetic to the reader. If you're not REALLY EXCITED (or if you've read any amount of Yudkowsky/Bostrom) you're probably not going to find a lot here.