Depending on how you define "battle", the struggle for Rabaul can be thought of as one continuous battle from early 1942 all the way to the end of the Pacific War.

From the initial invasion of Rabaul and destruction of Australia's Lark Force to the final steps of Operation Cartwheel to the Australian Army's return to Rabaul to isolate the Japanese garrison in 1945, there were daily strikes and counterstrikes around and above the island.

I can recommend Bruce Gamble's three book series on Rabaul (Invasion Rabaul, Fortress Rabaul and Target Rabaul) for a very in-depth look at the battle/campaign/whatever you want to call it.

Edit: One could also call the Battle of the Atlantic (1939 - 1945) one continuous campaign/battle too.

I think the siege of leningrad deserves a place in the discussion. About 4 yrs long