Operation Reindeer was when two German Alpine divisions, led by General Eduard Dietl attacked from the North Cape of Norway, through Finland, to Petsamo. They were supposed to continue on towards Murmansk. However, the Soviet's 14th and 52nd divisions stopped Dietl from capturing Petsamo. This battle was fought north of the Artic circle.
Unless you count the German battleship Tirpitz bombarding the coal mine on Spitzbergen island as a battle, this is the most northern battle ever fought. While Dietl failed to capture Petsamo or Murmansk, he was able to prevent the Soviets from invading Norway and getting west of the North Cape of Norway. These battles were fought along the shore of the Artic ocean.

When Finland switched sides in autumn 1944, the German 20. Gebirgsarmee under Dietl, who had been fighting around Petsamo in an attempt to reach Murmansk and cut that supply route for lend-leade aid was trapped in northern Finland.

This force, over the period of six months, attacked by Soviets and Finns, retreated over Finnmark (the northernmost Norwegian fylke/province) to Narvik, where they arrived in March and April 1945.

As /u/tungahost says, the Soviets stopped east of Varangerfjord, and the huge torched land became a no-mans-land.

In order to protect the territory from possible Soviet annexation and in order to help those made homeless by the Germans torching almost every building, Sweden allowed a battalion and two companies out of the Norwegian police (a term used to confuse intelligence, while they had police training, they were trained and equipped as regular Swedish infantry, with the exception of having British PIATs as well), in total 1 550 men were flown in on American Dakota transport planes from the airport in Kallax, Sweden, to Finnmark, to join the company of the Free Norwegian brigade in Britain which had landed at Murmansk with Soviet permission. These troops skirmished a bit with the retreating Germans with patrols as they attempted to disarm mines and booby traps set by the Germans.

This isn't a very large fight, but considerably further north than the ones mentioned by others in Scandanavia was a clash on Sabine Island in March of 1943.

The Germans had set up covert weather stations in Greenland, and tasked to track them down was "The Sledge Patrol", a small force of Danes, Norwegians and native Greenlanders, supported by the US Coast Guard who resupplied them periodically. On Sabine Island they had built a base camp, which was attacked by the Germans, resulting in one of the patrol's members, Eli Knudsen, being killed, the only man they lost in four years of work!

Another encounter, which resulted in no fatalities, came on October, 4, 1944, where a team of commandos were landed on Koldeway Island and captured a 12 man German weather team.

The book I'm using, 'War in the Outposts' by Simon Rigge, has a small section on the whole operation, but apparently there is an entire book called "The Sledge Patrol" if this sounds interesting too you. Looks very well reviewed in Amazon, but I haven't read it myself though.