The sequence of functions $f_0,f_1,f_2,...$ is given by the conditions:
$f_0(x) = |x|$ for all $x \in R$
$f_{n+1}(x) = |f_n(x)-2|$ for $n=0,1,2,...$ and all $x \in R$.
For each positive integer $n$, solve the equation $f_n(x)=1$.
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   - The solutions for \( f_n(x) = 1 \) are \( x \in \{ \pm (2k+1) \mid k \in \mathbb{Z}, |2k+1| \leq 2n+1 \} \).
   - This means \( x \) must be an odd integer and its magnitude is dependent on \( n \).

The final answer is \( \boxed{ x \in \{ \pm (2k+1) \mid k \in \mathbb{Z}, |2k+1| \leq 2n+1 \} } \).