There are 12 ruling monarchies in Europe. Of these, two are not of particular interest here, since they are not hereditary, and any relationships are likely to be untraceable and coincidental. These are Vatican City, ruled by Pope Francis (an elected position), and Andorra, which is a co-principality jointly held by the President of France (elected) and the Bishop of Urgell, the local diocese (appointed).
Of the ten remaining monarchies, the trick is to find the blood relative (common ancestor), rather than just ties by marriage (a relative). The closest living blood relative of all currently reigning European monarchs is John William Friso, Prince of Orange, who lived from 1687 to 1711. His relationships to the monarchs are as follows:
Belgium: King Philippe is both his 7x great grandson and his double-8x great grandson
Denmark: Queen Margrethe II is his 6x great granddaughter and his double 7x great granddaughter
Liechtenstein: Prince Hans-Adam II is his 7x great grandson
Luxembourg: Grand Duke Henri is his double 7x great grandson and his 8x great grandson
Monaco: Prince Albert II is his 8x great grandson
Netherlands: King Willem-Alexander is his 7x great grandson and double 8x great grandson
Norway: King Harald V is his 6x great grandson and double 7x great grandson
Spain: King Felipe VI is his triple 8x great grandson
Sweden: King Carl XVI is his 6x great grandson and triple 7x great grandson
United Kingdom: Queen Elizabeth II is his 6x great granddaughter
I never know quite how to cite an outright genealogy like this - it's not really a question of history in the usual way we think of it, but I dearly love the subject.
All of the current royal families in Europe - except the royal family of the Netherlands - are descended from Queen Victoria. The king of the Netherlands, while not descended from Queen Victoria, is descended from George II of England, an ancestor of Queen Victoria, so that would make George II the closest common ancestor of all European royalty.