We give a combinatorial construction of a one-parameter and a two-parameter family of complete caps in finite projective spaces over GF(2). As an application of our construction we find, for each \alpha in [1.89,2], a sequence of complete caps in PG(n,2) whose sizes grow roughly as \alpha^n . We also discuss the relevance of our caps to the problem of finding the least dependent caps of a given cardinality in a given dimension.