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The World's Oldest Fraggle's first appearance was in the Fraggle Rock episode The Finger of Light. As his name implies, he serves as elder for the Fraggles, often officiating at ceremonial events and emergency meetings. He often has to be corrected by his assistant Henchy Fraggle, to which he replies "I know that!" and whacks Henchy on the nose with his cane.
The World's Oldest Fraggle is obviously greatly respected by the Fraggles, but he's not their leader. His role seems to be mostly ceremonial; he presides over games and leads a meetings a lot, but he doesn't make rules for anyone. The Fraggles are their own leaders, and nobody is the boss.