Well, I was just wondering specifically which of the ten reigning monarchs can best make that claim. My instinctive feeling says that it's probably King Felipe VI of Spain, precisely because of the interfaith marriage of his parents (although his mother converted to Catholicism, she was raised Orthodox).
Queen Sofia has completely royal ancestry within at least three full generations (she actually has more blue blood than her husband, who has partial morganaut ancestry), and is related to all the Protestant and Orthodox royals of Europe. King Juan Carlos has nearly 100% royal ancestry within three full generations and is related to all the Catholic dynaties of Europe. In addition, he likely is related to almost all the Protestant dynasties as well (even the very Catholic Queen Maria Cristina of Spain had two Protestant grandmothers).
At the opposite end, I would imagine that Prince Albert II of Monaco is the least connected sovereign, with the weakest claim to being related to the entire Gotha. Prince Hans Adam II of Liechtenstein at least has direct ties to the Habsburg, Wittelsbachs, and Braganzas, thanks to his paternal grandmother (Archduchess Elisabeth Amalia of Austria).
Grand Duke Henri of Luxembourg probably comes close, but the Spanish king probably beats him by being descended from the Romanov czars of Russia. But he is still better connected than his Belgian cousin, King Philippe, whose mother (Queen Paola) doesn't really add anything, whereas the late Grand Duke Jean and Grand Duchess Josephine-Charlotte were a fully royal couple.
The Scandinavian monarchs all claim fully royal ancestry within at least three full generations, and probably don't boast extra connections than Her British Majesty, who has common ancestry through her mother and partially morganaut ancestry through her father. If anything, the Duke of Edinburgh is probably a more blue-blooded person than his wife ...
In any case, the British and Scandinavian reigning monarchs do not have close genealogical ties to Catholic or Orthodox royalty. King Willem IV Alexander of the Netherlands, however, is distantly descended from the Russian imperial dynasty, since Queen Anne (consort of King Willem II) was a daughter of Czar Paul. His ties to the Catholic royals, however, are rather remote.
So the Protestant sovereigns probably rank in the middle, when it comes to genealogical ties ...
Is all this correct?