A word that describes something impossible or near impossible to do but fun to try. One example I can think of is trying to lick one's elbow. A contortionist can do it but not everyone else. We sure give it a try despite knowing we can't do it.
Perhaps the word is not specific enough, but I'd simply suggest challenging.
Something challenging is demanding, so perhaps it doesn't quite stretch to being "near impossible", however a challenge suggests something which is "fun to try". I suppose it depends whether you are looking for an obscure or common word. HTH
You can use Conundrum or Enigma to refer to difficult but interesting problems.
As per google-
conundrum- confusing and difficult problem or question.
a question asked for amusement, typically one with a pun in its answer; a riddle.
Your endeavors are futile. Such a word simply does not exist.