Which of the following beers would enhance the flavors of a rich fatty meat?

The best choice for enhancing the flavors of rich fatty meat is a Barleywine. This style of beer has a high malt sweetness, robust body, and a higher alcohol content, which allows it to stand up to and complement the richness of fatty meats. The caramel and toffee notes present in Barleywine can harmonize beautifully with the savory, deep flavors of the meat, while the beer's sweetness helps to cut through the fat, creating a balanced taste experience.

In contrast, while amber lagers and Oktoberfest beers possess malt sweetness, they are lighter and may not have the necessary intensity or depth to enhance rich fatty meats effectively. Similarly, saisons are typically characterized by their dry, spicy, and fruity flavors, which might not provide the same level of complementary richness that Barleywine does with fatty meats. Thus, Barleywine is the choice that best enhances the overall dining experience with rich, fatty meat dishes.