Place the eggs into a saucepan in a single layer and fill with water to cover the eggs by 1 inch. Cover the saucepan and bring the water to a boil over high heat. Remove from the heat and let the eggs stand in the hot water for 15 minutes. Pour out the hot water; cool the eggs under cold running water in the sink. Peel and chop the cooled eggs.
Place the potatoes into a large pot and cover with water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until tender, 15 to 20 minutes. Drain and refrigerate until cold. Peel and cube once cold.
Stir together the mayonnaise, ranch dressing, relish, mustard, salt, pepper, paprika, and celery seed in a mixing bowl. Add the eggs, potatoes, onion, pepperoncini, and olives; stir until evenly mixed. Cover and refrigerate at least 2 hours before serving.