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There was no king of the House of Brunswick - ever. The highest ranking title was duke.
When Queen Anne died in 1714, there were many possible successors. However, by law they were excluded from the succession because they were Catholic, so the succession skipped to the 52nd in line to the British throne, a distant Protestant cousin George Louis of Brunswig-Luenenburg who became George I of England. George was the son of the Elector of Hanover (also Duke of Brunswig-Luenenburg) Ernest Augustus and his wife Sophia of the Palatinate. Sophia was a granddaughter of James VI [of Scotland] and I [of England]. Law had declared her the successor to Queen Anne since she was Protestant. She died two months before Anne, so the succession passed to her son George Louis.
The official name of the House of Hanover was House of Brunswig-Luenenburg, Hanover line. The Hanover line was a cadet branch of the House of Brunswig-Luenenburg. The House of Brunswig-Luenenburg became extinct in 1884 and only the House of Hanover survives as the remaining line of the House of Welf. The current head of the House of Hanover is Ernst August, Prince of Hanover.