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The Bonus Gauge is a game feature only available in Kingdom Hearts 358/2 Days, beginning on day 51. It multiplies one's mission rewards when it hits the indicated mark; the maximum multiplier is three. The Bonus Gauge is only available to certain missions, particularly those that appear at the same time as several others.
To receive the most benefits from the Bonus Gauge's multiplier, check the mission rewards for every mission where the Bonus Gauge is available and complete the missions which give the items needed last. By doing this, one can plan which set of items they will receive multiples of.