This olive oil cake from Frankies Spuntino in Brooklyn was one of the nicest things I ate in NYC...oily with a hint of orange.
I didn't have a bundt tin, so it was a bit oily in the middle, but still delicious (took longer to cook and so the edges were crusty). A lot of olive oil, but it's worth getting one that tastes nice... :

5 large eggs, at room temperature
finely grated zest of 1 1/2 oranges
1 1/4 cups sugar
2 cups extra-virgin olive oil (you could get away with less)
2 cups cake flour
1 1/8 teaspoons baking powder
1 teaspoon fine sea salt

Preheat the oven to 325°F and butter and flour a 10-cup bundt pan. In a bowl, using a handheld electric mixer, beat the eggs with the sugar and orange zest at medium-high speed until smooth. Gradually beat in the olive oil until creamy, about 2 minutes. In a small bowl, whisk the cake flour with the baking powder and salt. Add the dry ingredients to the egg mixture in 3 batches, beating on medium speed between additions. Scrape the batter into the prepared pan and bake in the center of the oven for about 1 hour, until a toothpick inserted in the center of the cake comes out clean. Let cool in the pan for 15 minutes, then invert onto a rack. Let the cake cool completely before cutting into slices and serving.