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The example of emergence that comes to my mind most readily is a simple observation that Douglas Hofstadter made in Godel, Escher, Bach -- a book which definitely does not use "emergent" as a synonym for "magical":
In a game of Go, once there are two separate open spaces -- "eyes" -- in the middle of a connected group of stones, that group becomes invincible (because the opponent can't fill both holes with one move). There's no official rule in Go that says "Patterns with two eyes can't be captured", the rule just says that to capture a group you have to surround it completely and leave no open spaces. Thus two-eye invincibility is an emergent consequence of the rules of Go.
It's important that this new emergent rule is a significant simplification: once you realize that two eyes are invincible, you no longer have to do any complicated analysis about how close a two-eyed group is to being completely surrounded. It's safe, full stop (at least as long as you don't stupidly fill in the holes yourself).
The game of Go has very few rules. In practice, the two-eye invincibility "rule" is a very important and useful one, if you want to play the game well. To try to force Eliezer to talk about "emergence" or something equivalent, I would ask: where did the two-eye invincibility rule come from?
-- Okay, now, so what isn't emergent? There's a another Go rule, the "ko" rule, which says you can't play in such a way as to get the exact same board position back after two moves: no capture followed by immediate recapture unless it changes the board. There's nothing "emergent" about that rule that I can think of -- it helps keep Go games from going on forever, but it has no simple-but-unexpected high-level consequences.
There are a lot of strategic patterns of play in Go that are not emergent, either -- e.g., there are no simple rules of thumb that can tell you, in all cases, whether a group with one eye or no eyes can be captured or not. Often the answer depends on a single apparently unrelated stone way over on the other side of the board. No simplifications available here, therefore no emergence.
There are many other more involved examples of emergence (and non-emergence) -- gliders and spaceships in Conway's Game of Life come to mind, and Herschels and random ash densities -- but this blog comment is too narrow to contain a good summary of them all...
Two other books that do a fine job (in my opinion) of describing the concept of "emergence" as distinct from "magic" are Cohen and Stewart's The Collapse of Chaos and Figments of Reality.