Example 6 How many positive real numbers $x$ satisfy the equation $x[x[x[x]]]=2006$?
Assume there exists a positive real number $x$ satisfying $x[x[x[x]]]=2006$.
If $02006$.
From this, we know $6<x<7,[x]=6$.
Thus, $2006=x[x[6 x]]$.
But from $x<7$, we know $6 x<42$, so $[6 x] \leqslant 41$.
Therefore, $x[6 x]<7 \times 41=287$.
So, $[x[6 x]] \leqslant 286$.
Hence, $x[x[6 x]]<7 \times 286=2002<2006$, contradiction!
Therefore, the number of positive real numbers $x$ that satisfy the equation $x[x[x[x]]]=2006$ is 0.