Battleships, far from being a ridiculous idea, were a logical and necessary step for any navy wishing to project power at sea. They were so heavily armoured and gunned that for a long time, from the late 19th Century to just before WWII, the only realistic defence against a battleship was another battleship.
Or, playing the "wolf card" amidst the fog of war and a resourcefully laid smokescreen with a bunch of smaller ships. (Causing the enemy commander to think, "They must have larger ships back there." This was destroyers vs. cruisers, not battleships, though.)
http://en.wikipedia.org/wiki/Battle_off_Samar
I suspect that the future belongs to drones. Autonomous Air Independent Propulsion submersible drones would be far less expensive than an aircraft carrier, but very likely to be able to lie in wait for a carrier and torpedo it.
EDIT: Correction -- one could actually say those American destroyers engaged the Yamato, which was the biggest, baddest battleship ever.