The water level will in fact go up, since the anchor is more dense that water. Basic Archimedes stuff. Before the anchor was dropped the floating boat experienced an upthrust equal to the weight of water displaced.
And if you all like this sort of thing, try this:
A jet transport is in the cruise. Autothrottle and autopilot are not engaged and the aeroplane is in straight and level flight at constant speed and trimmed.
This aeroplane has no seats installed. Instead it has 1000 bird perches fixed to the floor and on each perch sits a parrot. A human being claps hands loudly and all 1000 parrots get airborne and fly around the cabin.
Question:
Does the aeroplane stay level, climb, or descend?
This question was part of one of my Central Flying School oral examinations.