noun
unbelievably; incredibly; to an unbelievable extent
Used adverbially to emphasize an extreme degree, often with adjectives or verbs. Literally 'to the extent that it's impossible to exist/be true'.
彼はありえないほど速く走る。
He runs unbelievably fast.
その映画はありえないほど面白かった。
That movie was incredibly interesting.
Similar meaning 'unbelievably', but ありえないほど carries a stronger nuance of impossibility or absurdity, while 信じられないほど is more about being hard to believe.
From あり得ない (arienai, 'impossible, can't exist') + ほど (hodo, 'extent, degree'). The phrase is a fixed adverbial expression.