In OTL Prussia owned several of the islands in the Virgin Islands (in both modern-day US and British) in the late 1600s and early 1700s, if Prussia could be more successful with their small islands than the Dutch, French, and British were with their smaller islands, we could see Prussia able to buy more of the islands that drain the resources of the other European nations; but will never get big ones like Guadeloupe or Martinique for instance, although- Sweden received Guadeloupe from the Napoleonic wars and returned it to France in return for quite a big amount of money, we could see it given at the peace treaty to Prussia instead or Sweden sold it to Prussia for more money. With enough of a mass of small island colonies, it is possible that Prussia preempts Napoleon's selling Louisiana and gets Nappy to sell it to them instead in return for supporting him, or in one of the times when they oppose Nappy they may try to seize it (especially in a seizing scenario we may see the US take St Louis and then buy everything north of OTL Louisiana and let the Prussians have New Orleans and OTL Louisiana). In this scenario we could see the Monroe Doctrine harder to enforce and Prussia may seize weakly defended portions of weak nations (most likely Texas) and maybe interfere to help places like Yucatan stay independent as a protectorate. Prussia/Germany may even be the one to build the Panama Canal or one in Nicaragua.