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There is most certainly not any fountain in Dark Dawn... In fact I can't think of any mini-games at all. Does avoiding the light at the Apollo Lens with the Umbra Gear count? That is the only thing I can think of.
I noticed that there was a lack a minigames during my first playthrough. I was so intrigued by the newness and story, and blazed through the easy difficulty, that I didn't really care about it. But I do miss those though; especially the slots game where you could get Mythril Shirts and Hyper Boots.