Put water in large mixing bowl. Sprinkle yeast on water. Beat in honey, olive oil, 2 eggs and salt with a wire whisk. Add flour one cup at a time, whisking after each addition. Begin using a wooden spoon, then a floured hand as your dough thickens. Knead the dough in the bowl for a few minutes, until it is smooth, elastic and no longer sticky. Transfer the dough to a floured surface to prevent sticking, then lightly oil the bowl. Place the dough back in the bowl, cover with a cloth and set it in a warm place to rise, about 1 1/2 hours or until the dough doubles in size.
Punch down dough and turn onto a floured surface. Divide in half and knead each half for 5 minutes, adding flour as needed if it feels too sticky. Divide each half into thirds and roll each piece into a long snake 1 1/2 inches in diameter. Line up 2 sets of 3 snakes and then braid each from the middle. Press the ends firmly together.
Lightly oil 2 baking sheets. Place one braided dough on each. Cover with a towel and let rise for one more hour, or until doubled in size again.
Preheat oven to 375°. Beat remaining egg in a bowl and brush a generous amount over each braid. Bake for about 40 minutes. Remove loaves from sheets immediately and cool on a rack for at least 30 minutes before slicing.