in the first part of the talk, of elementary nature, we review some problems that are of interest to division algebra specialists, among which the constructions of the so-called indecomposable and non-crossed products division algebras (the two exotic specimens alluded in the title). Next, we outline their constructions over $Q_p(t)$ (and finite extensions thereof), focusing on indecomposable ones; this is joint work with K. McKinnie and E. Brussel.