Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$.
1. Let \( p(x) \) be a polynomial of degree strictly less than \( 100 \) and such that it does not have \( (x^3 - x) \) as a factor. We need to find the smallest possible degree of \( f(x) \) in the expression:
   \[
   \frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3 - x}\right) = \frac{f(x)}{g(x)}
   \]
   for some polynomials \( f(x) \) and \( g(x) \).

2. Since \( x(x^2 - 1) \nmid p(x) \), we can write:
   \[
   p(x) = x(x^2 - 1)P_1(x) + P_2(x)
   \]
   where \( \deg(P_2(x)) < 3 \).

3. Then:
   \[
   \frac{p(x)}{x^3 - x} = P_1(x) + \frac{P_2(x)}{x^3 - x}
   \]

4. Taking the 100th derivative:
   \[
   \left(\frac{p(x)}{x^3 - x}\right)^{(100)} = P_1^{(100)}(x) + \left(\frac{P_2(x)}{x^3 - x}\right)^{(100)}
   \]

5. Since \( \deg(P(x)) < 100 \implies \deg(P_1(x)) < 97 \), it follows that \( P_1^{(100)}(x) = 0 \).

6. To minimize the degree of \( f(x) \), we must have \( P_2(x) = k \) for some constant \( k \). Thus:
   \[
   \left(\frac{p(x)}{x^3 - x}\right)^{(100)} = \left(\frac{k}{x^3 - x}\right)^{(100)}
   \]

7. We claim that:
   \[
   \left(\frac{k}{x^3 - x}\right)^{(n)} = \frac{f(x)}{(x^3 - x)^{n+1}}
   \]
   and \( \deg(f(x)) = 2n \) for any \( n \in \mathbb{N} \).

8. Proof by induction:
   - Base case: For \( n = 1 \) and \( n = 2 \), the claim holds.
   - Inductive step: Assume the claim holds for \( n = c \). Then:
     \[
     \left(\frac{k}{x^3 - x}\right)^{(c+1)} = \frac{f'(x)(x^3 - x)^{c+1} - (c+1)(x^3 - x)^c (3x^2 - 1) f(x)}{(x^3 - x)^{2(c+1)}}
     \]
     Simplifying, we get:
     \[
     \left(\frac{k}{x^3 - x}\right)^{(c+1)} = \frac{f'(x)(x^3 - x) - (c+1)(3x^2 - 1) f(x)}{(x^3 - x)^{c+2}}
     \]
     Since for the \( c \)-th derivative, \( \deg(f(x)) = 2c \), for the \( (c+1) \)-th derivative, \( \deg(f(x)) = \deg(f'(x)(x^3 - x)) = 2(c+1) \).

9. By the principle of mathematical induction, the claim is true.

10. Therefore, for the 100th derivative, the minimum degree of \( f(x) \) is:
    \[
    \boxed{200}
    \]