4. Castles of Burgundy
Interesting enough, this is the one game on my list that I do not personally own. Regardless, I still get to play it enough with others that do have this game.
This game really is a Stefan Feld masterpiece. There are so many options, so many ways to formulate strategies for your turns, and still plenty of player interaction and friendly competition. The mechanic of rolling your two dice on each turn, possibly manipulating the result with additional workers, and then using the results to conduct actions is great at capturing the interesting aspect of randomness while also keeping the game quite balanced and leaving the player with the choice of how much control and security he or she wants to have (at the sacrifice of a few actions).
Additionally, there is the enjoyable aspect of building your own kingdom. Guaranteed, this kind of building is more strategic than thematic (as it typically is with a Euro game), but the general idea still remains.
In short, this is a model Euro game that I would recommend for anyone who wants to explore beyond the gateway board games (i.e. Settlers of Catan, Ticket to Ride, etc.).