This paper proposes a novel Adaptive Super-Twisting Algorithm (STA) to cope with bounded and unbounded disturbances. The proposed adaptive multilayered scheme does not require the knowledge of upper bounds for the matched disturbances and their derivatives. The only requirement is that the m-th derivative is bounded by an unknown constant. Using a modified low-pass filter we can estimate the disturbance and its derivatives so that both gains of the STA can be adapted in a non-conservative way, enforcing a second-order sliding mode and reducing the undesirable effects of chattering.