Recall that the n-solvable filtration of the smooth knot concordance group, suggested by work of Cochran-Orr-Teichner, has the property that any topologically slice knot lies in every term of the filtration. With the aim of studing topologically slice knots, we investigate a new filtration, , that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. As is the case for the n-solvable filtration, each has infinite rank. But our primary interest is in the induced filtration, , on the subgroup, , of knots that are topologically slice. We prove that is large, detected by gauge-theoretic invariants and the , , and -invariants; while the non-triviality of can be detected by certain -invariants. All of these concordance obstructions vanish for knots in . Nonetheless, going beyond this, our main result is that has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.