There is no such number. Like Ross mentioned in his answer, this is related to Richard's paradox.
At the heart of both this puzzle and Richard's paradox is (from the Wikipedia article)
the resolution of Richard's paradox is that there is no way to unambiguously determine exactly which English sentences are definitions of real numbers
The resolution of this puzzle is that although it sure seems like we are defining a real number, we are actually not defining one. For example, consider the following statement:
The first natural number that is greater than five and less than three.
This obviously does not describe a number as no matter which number you try one of the two requirements will always be left unsatisfied. Likewise, the statement
The first natural number, which can not be defined with fewer than 15 words.
has a requirement that cannot be met - it must not be able to be defined with fewer than 15 words, and yet we have done so. Thus, there is no such number. The only real difference between the two statements is in how obvious the unsatisfiable requirements are.