Why do highly branched alkanes have higher octane numbers than their corresponding linear isomer?
The reaction desired in an automobile piston is a rapid burning, not an explosion (very very rapid reaction). If the air + gasoline vapor burns much faster than the camshaft is turning, the explosion causes a ping, or high pressure rise in a smaller volume (because the piston has not moved far enough). This can damage the piston, or at least add an unusual stress.

Burning is a radical reaction and the radicals produced from linear alkanes are quite reactive (i.e., fast reactors). Branched alkanes might lose a hydrogen, even break apart, but the free electron is more stable, and the reaction rate is slower. 

There are compounds with a higher octane number than iso-octane. E.g., toluene has an octane number ~120, and was highly sought after for airplane fuel during WW II. The benzylic radical which is the likely first product of a radical reaction is quite stable (but not inert).