In this paper we determine, for all (Formula presented.) sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting (Formula presented.) -tuples of nilpotent elements of (Formula presented.). Our main result is that in characteristic (Formula presented.), this nilpotent commuting variety has dimension (Formula presented.) for (Formula presented.), (Formula presented.). We use this to find the dimension of the (ordinary) (Formula presented.) th commuting varieties of (Formula presented.) and (Formula presented.) for the same range of values of (Formula presented.) and (Formula presented.). Our principal motivation is the connection between nilpotent commuting varieties and cohomological complexity of finite group schemes, which we exploit in the last section of the paper to obtain explicit values for complexities of a large family of modules over the (Formula presented.) th Frobenius kernel (Formula presented.). These results indicate an inequality between the complexities of a rational (Formula presented.) -module (Formula presented.) when restricted to (Formula presented.) or to (Formula presented.); we subsequently establish this inequality for every simple algebraic group (Formula presented.) defined over an algebraically closed field of good characteristic, significantly extending the main theorem of Lin and Nakano, Inventiones Mathematicae, 138 (1999), 85–101.