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I suppose it's theoretically possible to produce a set of fish that could do what you're asking, but I don't know of anyone in this community that has attempted to determine what those fish would be. That set of fish would have to contain fish from several iterations of breeding (i.e. generations), or as they are referred to in the game: common, unusual, and rare fish. Most people who opt to be structured in their breeding (as opposed to relying on random breeding and luck) have found that there are a handful of fishies that will allow you to breed all other species, but in multiple generations, rather than one.