A gaming device with an accumulator that accumulates re-triggering symbols until a pre-determined number of re-triggering symbols are obtained by the player. A variable amount of free activations as well as a variable threshold amount of re-triggering symbols are necessary to re-trigger a secondary game. The gaming device provides a plurality of player selectable selections. Each selection is associated with a number of activations of the symbol generators as well as a number of re-triggering symbols necessary to re-trigger the secondary game. If the player picked threshold of re-triggering the secondary game is satisfied (i.e., the required amount of re-triggering symbols associated with the player picked selection were accumulated during the provided amount of spins associated with the player picked selection), the gaming device subsequently re-triggers the secondary game and again provides the player a plurality of selectable selections which are each associated with a number of spins of the reels as well as a number of accumulator symbols necessary to re-trigger the secondary game.