While I love mythological beasts, particularly researching the origins of various creatures, I am certainly not under the delusion that I am one of them.
Also, even if I were to pick an option on your poll: Dragon myths are far too wide-spread and vague (ranging from the legless, wingless wyrms that were said to curl round medieval hills to the flying, fire-breathed monster St. George is said to have slain to the less dangerous, water-associated lung of China to the Norse myth of Jormugard, the Serpent of the World) to allow anyone versed in much of them to choose easily. Unicorns (the vicious Persian kar-kar-dahn, the amazing conglomerate of many beasts from heraldry, the more peaceful Chinese ki-lin) have a similar problem. 'Faery' can refer to any member of the Unseelie or Seelie Courts, which ranges from kelpies to red-caps to brownies, but when used by itself usually refers to stories of changelings--not something I'd choose to be associated with. Narrow it down a bit? Or are we talking those hideously-watered down things you see slapped onto Barbie packages and the most saccharine of children's fantasy novels?
If I was to choose a favorite mythical beast, it would be the poor, forgotten catoblepas.
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cogito, ergo sum.
non cogitas, ergo non es.