Currently Online 
The genre and this series in general usually gets a lot of flack from gamers, but 'Call of Duty' is a real classic, shaping the modern first-person shooter for generations to come. Meant to be just an ordinary military shooter, it unexpectedly (or maybe it was just the terrific gameplay design) rose far beyond that, and still has a very much active online community.
Even though this is one of the few games I actually played online extensively, the game shines bright in it's single-player mode. The story follows the diary entries of three Allied soldiers during World War II, an Englishman, an American and a Soviet Russian, each soldier in turn and ending with the Russian soldier and the final battle to take the Reichstag in Berlin. Both these soldiers, their experiences and the missions are semi-based on real-life accounts. The add-on 'United Offensive' added even more missions, among them some that are taking place during the horrifying Battle of the Bulge in the Ardennes.
Taking place during World War II meant the game already has a massive advantage to me compared to most other shooters, but it also is just absolutely great in terms of solid gameplay and good, engaging level-design. In reality, one might swap this game in this list for the second installment, which is actually just as good and is a clean continuation of the foundation set here, but loses on account of the novelty factor.
