Some of the really great numbers are
transcendental. Rational numbers can be expressed as a ratio (finite fraction), irrationals cannot, such as the square root of 2 or the Golden Mean. These require the use of polynomial roots. But the transcendentals can't be expressed exactly with roots--these include numbers like pi and "e."
I think (js1138) mentioned Steven Weinberg. I think he is also quoted as saying something like "the more we learn about the universe, the more pointless it all seems."