In this paper we give examples of skew-product maps T : T-2 -> T-2 of the form T (x, y) = (x + omega, x + f(y)), where f : T -> T is an explicit C-1-endomorphism of degree two with a unique critical point and omega belongs to a set of positive measure, for which the fibered Lyapunov exponent is positive for a.e. (x, y) is an element of T-2. The critical point is of type f '(+/- e(-epsilon)) approximate to e(-beta s/(ln s)2) for all large s, where beta > 0 is a small numerical constant.