Simpson proved that every countable model of PA has an expansion (to PA*) that is pointwise definable. The natural question, then, is if every countable model has an expansion to PA* in which no new elements are defined. Enayat proved this is false by showing the existence of many models that are not pointwise definable, but become so upon addition of any undefinable class. Inspired by this, I have begun thinking about which models have this property. I will describe some models with this property (and some without) and talk about my search for a non-trivial such model (I will also explain what I mean by “non-trivial” here).