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The inductor stores energy in magnetic field. Since the energy cannot be immediately transferred to or withdrawn from the magnetic field, some delay is inevitably associated with current increase/decrease in any inductor.
I might ask "Why can't energy be transferred/withdrawn immediately - at least at the speed of light?" I think you should expand this answer by saying the rising current causes a rising magnetic field (changing flux) in each loop of the inductor. This changing flux gives rise to an emf (Faraday's Law) which gives rise to a current in a direction to oppose the change in magnetic field (Lenz's Law). The direction of that current is opposite to the currect from the outside source, reducing the instantaneous net current ... thus the delay in the rise of the current through the inductor to its final value...