If I cast a spell with X in its mana cost "without paying its mana cost" what is the value of X?
The value of X is chosen when the spell is cast.
But the only legal choice for X is zero in this case.


  107.3b If a player is casting a spell that has an {X} in its mana cost, the value of X isn’t defined by the text of that spell, and an effect lets that player cast that spell while paying neither its mana cost nor an alternative cost that includes X, then the only legal choice for X is 0. This doesn’t apply to effects that only reduce a cost, even if they reduce it to zero. See rule 601, “Casting Spells.”


This is confirmed by a ruling on Omnispell Adept.


  If a spell has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.