A nice example of seemingly trivial structure that hides highly nontrivial structure
is that of a projective space. Such a space consists of "points', "lines", and "planes"
with the obvious properties: there is a unique line through any two points, any two
planes meet in a unique line, three points not on a line lie on a unique plane,
and so on.

Surprisingly, any such space has an underlying *skew field* which coordinatizes
the space so that lines and planes have linear equations.  This due to the fact 
that the Desargues theorem holds in any projective space. Hilbert (1899) showed (in a highly roundabout way) that one can then define sum and product of
points, and use the Desargues theorem to prove their skew field properties.