1. A gaming system comprising: at least one input device; at least one display device; at least one processor; and at least one memory device storing a plurality of instructions which, when executed by the at least one processor, cause the at least one processor to operate with the at least one display device and the at least one input device to: (a) display a non-progressive award associated with a secondary game, the non-progressive award having a nonzero value; (b) display a play of a base game; (c) upon an occurrence of a designated award display changing triggering event: (i) remove the display of the non-progressive award, and (ii) display a progressive award in place of the non-progressive award, said progressive award being associated with the secondary game; and (d) upon an occurrence of a secondary game triggering event, display a play of the secondary game and display any awards associated with said play of the secondary game, wherein: (i) if the non-progressive award is displayed, the non-progressive award is winnable in said play of the secondary game; and (ii) if the progressive award is displayed, the progressive award is winnable in said play of the secondary game.