How does the number of eggs affect a cookie recipe?
Eggs contribute to better texture, leavening and they extend shelf life. More eggs = moister (sp!) cookie. Eggs are also crucial in building structure. They are about 75% moisture, 12% protein, 10% fat and 2ish % sugar.

The white provides strength, stability and moisture.

Yolks, where all of the fat is in an egg, increase richness, tenderness and flavor.

Therefore, if you put an extra egg, you will get a chewier cookie. I do it all the time. If you put less, you will get a more crumbly cookie.