Is it possible to design cascaded LC filters with inductors sharing the same core?
No, this will not give you what you want because all the inductors will be coupled at nearly 98%. Any voltage across L1 will be mirrored across L2, L3 etc..

The filter topology shown is a low pass type so I'm assuming the power signal you wish to remove is higher than what you wish to pass. The higher frequency power signal will, due to the first filter capacitor being a low impedance to that frequency, appear largely across L1 and either inverted or of the same phase across L2 and L3. No matter what you do, with near 100% coupling and the capacitors being low impedance to the power signal you wish to remove, the final inductor will produce virtually the same power signal at the output as seen at the input minus a few percent due to winding losses etc..