The smallest example of more than the sum
Take two hydrogen atoms (gas) and one oxygen atom (gas). Combine them and you get H₂O. Liquid water. Drinkable, life-sustaining, everywhere. None of water's properties live in H or O alone. Hydrogen and oxygen, separately, would not let you drink them, swim in them, or grow life from them. The properties show up only at the molecular layer. They emerge.
This is the first time the word emergence earns its weight in this quest. New properties appear at a new layer that aren't predictable from the components individually. Water's properties — its strange density curve (ice less dense than liquid, which is why ice floats and lakes don't freeze solid), its surface tension, its solvent behavior — are layer-2 properties. The atoms underneath them don't have those properties.
Why ice floats — a small mystery you've been swimming in
Almost every substance is denser as a solid than as a liquid. Water is one of the rare exceptions. Ice floats. If it didn't, lakes would freeze from the bottom up, killing aquatic life every winter. We'd live on a much colder, deader planet.
The reason: water molecules form hydrogen bonds — weak attractions between the slightly-positive H and slightly-negative O of neighboring molecules. As water cools, those bonds organize the molecules into a hexagonal lattice with empty space inside. Solid ice is less dense than the liquid because of the spacing in the lattice. A molecular property (the geometry of H-bonds) creates a planetary-scale property (lakes that don't freeze solid). Layers of emergence stacked on layers.
The general shape
This layering doesn't stop with water. Atoms emerge from particles. Molecules emerge from atoms. Cells emerge from molecules. Tissues from cells. Organisms from tissues. Societies from organisms. Each layer has properties that don't exist below it. Each layer is, in a real sense, a new world that the lower layer didn't know it was building.
This is the spine of the whole quest. Particles → atoms → molecules → life → consciousness → AI → us. 1+1=2+emergence. Every layer adds something the previous layer didn't promise.
The next lessons in counting (universe alphabet, life-and-position) make the same point at different scales. The track ends when you can see emergent layers stacked vertically through everything around you.