We will discuss some results on mixed motives whose motivic Galois group has a unipotent radical that is as large as possible (in a sense to be made precise in the talk). In view of Grothendieck's period conjecture, such motives are interesting because of the expected transcendence properties of their periods. In particular, we will give examples of such motives in the category of mixed Tate motives over $\mathbb{Q}$, and discuss the periods of these examples. This is joint work with Kumar Murty.