It is definitely possible. There may be zoning restrictions that would prohibit it, or covenants in a homeowner's association. You may also be able to argue easement rights to the open space above your neighbor's home. A creative real estate attorney might be able to help you. I would suggest making an appointment and getting a consultation with an experienced real property attorney in your area.
*This answer is intended for informational purposes only and does not establish an attorney-client relationship or give rise to any privilege.
I agree with the previous answer. A real estate attorney can help evaluate the recorded documents related to the properties to ascertain whether your neighbor is permitted to build.
They can build up to the applicable height restriction which is generally 26 feet in Seatle single family zoning measured from the highest spot on the lot. You could offer to buy a view easement where they put a deed restriction on their lot and then you would have a protected view that would stay with your lot foreever and increase the value of your lot - and decrease the value of the neighbor's lot so assume it will be expensive. If they need a variance ( permisison from the city) to exceed the height restriction to do the 3rd story, you will get noitce and can fight it. If what they are proposing to do sort of looks like what other houses have done in the neighborhood, they probably are allowed to it. Can you build up on your house also?