A gaming device having a bonus scheme wherein an accumulator accumulates bonus re-triggering symbols until a pre-determined number of bonus re-triggering symbols are obtained by the player in the bonus game. The accumulated bonus re-triggering symbols are preferably the same symbol. In one embodiment, the bonus re-triggering symbols in the bonus game are the same symbol as the bonus triggering symbols in the primary game. In another embodiment, the bonus re-triggering symbols in the bonus game are different symbols than the bonus trigger symbols in the primary game. The bonus scheme of the present invention is preferably employed in conjunction with one or more other re-trigger bonus schemes. In another embodiment, the accumulator is employed as a bonus re-triggering event in the bonus game.