I will discuss the growth rate of the number of elements of the mapping class group of each Nielsen-Thurston type, that is, either finite-order, reducible, or pseudo-Anosov. We approach this question from the perspective of the so-called lattice point counting problem for the mapping class group acting on Teichmuller space, which concerns the number of group elements that send a given point into a ball of radius R about another point. In the case of a closed surface of genus g, Athreya, Bufetov, Eskin, and Mirzakhani have shown that, for the whole mapping class group, this quantity is asymptotic to exp((6g-6)R) as R tends to infinity. Maher has obtained the same asymptotics for those orbit points that are translates by pseudo-Anosov elements. We consider the remaining Nielsen-Thurston types: For finite-order elements we show the associated count grows coarsely at the rate of exp((3g-3)R), that is with exactly half the exponent, and for reducible elements it grows at the rate exp((6g-7)R). In order to achieve this, we introduce a new notion of "complexity length" in Teichmuller space which has several interesting features reflecting aspects of negative curvature. Joint work with Howard Masur.