Uh, I think of an ASBish sort of scenario:
- U.S. is generally more successful in purchasing/conquering former Spanish Caribbean-states during the collapse of the Spanish Empire.
- U.S. senate does not approve the sale of Alaska and Russia does not want to sell it with Britain, they hold out for awhile until there is the discovery of Gold there.
- Assume history goes similar enough to OTL (i.e. Franco-Prussian War of sorts, continued colonialism, etc.)
- Some analogue of WWI happens, only an Anglo-German alliance forms against a Franco-Russian bloc. The peace is somewhat like Versailles but against France. U.S. stays neutral.
- France devolves into far-right wing, Fascist-type state (perhaps with a puppet Monarchy that was restored after WWI? I don't know). Eventually invades Germany.
- France threatens neutral U.S. shipping and uses a newly built navy to attack U.S. ships. It's mostly pathetic, but let's say some over-excited saboteurs blow up a luxury liner in Caribbean waters that had a few U.S. senators going to gamble in Cuba or something. The U.S. demands the French allow U.S. ships under any flag free passage to Europe, they reject, etc. relations crumble and the U.S. launches a massive naval invasion of all French possessions in North America justified under the Monroe Doctrine (well not really but the U.S. gov't is more than likely to be outraged). Occupation continues after the war when the United States "pledges" to make each state an independent republic after the war, but the election of a new gov't in 1944 is far more 'gung-ho America' and pushes for the annexation of the islands.
- U.S. eventually joins in on the war, using Greenland as a base for forward shipping against Nazi France u-boat/some dangerous ship attacks. At the end of the war, the U.S. negotiates with Denmark to buy Greenland for several million dollars.
There. Fill in the details, but it's roughly a rip off of OTL history spiced with nice parts that get your job done. If anything at least it's mildly entertaining.
But in all honesty, it's much easier for the U.S. to simply influence the Caribbean states and own military bases on them, while owning Hawai'i because of its significant geopolitical and economical location. Owning Greenland however, makes sense and IOTL the U.S. did offer to purchase it at the end of WWII, but Denmark didn't take it.