If (R, m) is local and N is a non-zero finitely generated R-module of dimension n>0, then the top degree of its local cohomology module at the maximal ideal is not finitely generated. On the other hand, H^{0} as a submodule of N, is finitely generated. We investigate the question the least integer i such that H^{i}_{m}(N) is not finitely generated, an invariant known as the finiteness dimension.