1. A gaming system comprising: a gaming device including at least one input device; at least one display device associated with said gaming device; and at least one controller configured to operate with said gaming device and said at least one display device to: (a) associate one of a plurality of different virtual competitors with a player; (b) accumulate a first quantity of eligibility points for the associated competitor for an eligibility point accumulation event which occurs in association with at least one play of a primary game on said gaming device at a gaming establishment; (c) accumulate a second quantity of eligibility points for the associated competitor based on at least one of a plurality of different inputs made by the player remote from any gaming establishment; (d) form a total quantity of eligibility points based on the first quantity of eligibility points and the second quantity of eligibility points; and (e) upon a competition sequence triggering event, determine whether the associated competitor is eligible to participate in a competition sequence, said determination based on said total quantity of eligibility points.