Let S be a surface of genus $g\geq 2$. We show that for every even number $h\leq 6g-6$, there are infinitely many conjugacy classes of pseudo-Anosov mapping classes whose stretch factor is an algebraic integer in a totally real number field of degree h over $\mathbb{Q}$, with maximal Galois group. If time permits we discuss some applications to the geometry of moduli space.