  A proper elementary extension of a model is called small if it realizes no
new types over any finite set in the base model. We answer a question of
Marker, and show that it is possible to have an o-minimal structure with a
maximal small extension. Our construction yields such a structure for any
cardinality. We show that in some cases, notably when the base structure is
countable, the maximal small extension has maximal possible cardinality.
