An Infinitely Nested Radical

Number Theory Level 3

It can be shown that for any positive integer $n$ , the infinitely nested radical expression $\sqrt{ n + \sqrt{n + \sqrt{n + \cdots}}}$ equals a finite number. What is the largest positive integer $n \le 999$ such that this expression is equal to a positive integer?

Details and assumptions:

A nested radical expression is one which contains a radical inside another, as in $\sqrt{ 3 + \sqrt{5}}.$
An infinitely nested radical expression is one in which the radicals continue to an infinite extent.