We employ the Hrushovski Amalgamation Construction to generate strongly minimal examples of interesting recursive model theoretic phenomena. We show that there exists a strongly minimal theory whose only recursively presentable models are prime or saturated. We show that there exists a strongly minimal theory in a language with finite signature whose only recursively presentable model is the saturated model. Similarly, we show that for every $k \in \omega+1$ there exists a strongly minimal theory in a language with finite signature whose recursively presentable models are those with dimension less than $k$. Finally, we characterize the complexity of strongly minimal or $\aleph_0$-categorical theories that have only recursively presentable models by generating examples in every possible tt-degree.