If I'm not wrong, some years ago AA was interested in establishing a JFK-TRN flight (don't know if on daily or weekly basis), but seems that Alitalia put a big pressure to avoid it (and won, obviously... ).
"Dia-Oh, yeah, duh! How could I possibly forget about Berlin...now why does Berlin not have transatlantic service when it seems like a half-dozen cities in Germany that are smaller do..."
The only German cities with transatlantic services are FRA, MUC, STR and DUS (LH/Privatair Business Jets only). HAJ has a charter to Canada in the summer once or twice a week but I don't think that counts.
The reason why Berlin doesn't have any has been discussed quite often.
There is not enough demand. LH tried it in 2000/2001 with TXL-IAD but canceled it right away after 9/11 as it was performing very poorly.
In response to the last post I'm not sure if American ever wanted to start JFK-TRN, but I know in 1994 that TWA was about to begin this route with 4 weekly flights but nothing came of it.
The largest cities without service by U.S. carriers would probably be St. Peterburg with a population of 4.7 million and Berlin with about 3.7 million.
There are several other important cities in Europe with no U.S. carriers serving them.