Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$:
\[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\]
For which $k$ does there exist a $k$-tastrophic function?
1. **Define the function and the problem:**
   We need to find for which integer \( k > 1 \) there exists a \( k \)-tastrophic function \( f: \mathbb{N^*} \to \mathbb{N^*} \) such that for every integer \( n > 0 \), the \( k \)-th iteration of \( f \), denoted \( f_k(n) \), satisfies:
   \[
   f_k(n) = n^k
   \]
   where \( f_k \) is defined as:
   \[
   f_k(n) = \underbrace{f \circ f \circ \cdots \circ f}_{k \text{ times}}(n)
   \]

2. **Introduce the concept of \( v_k(n) \):**
   Let \( v_k(n) \) be the greatest power of \( k \) which divides \( n \). This means \( v_k(n) \) is the largest integer \( m \) such that \( k^m \) divides \( n \).

3. **Define the set \( A \):**
   Let \( A \) be the set of all positive integers which are not divisible by \( k \). Since \( k > 1 \), \( A \) is an infinite set.

4. **Partition the set \( A \):**
   Split \( A \) into \( k \) equinumerous subsets \( A_1, A_2, \ldots, A_k \). This means each subset \( A_i \) has the same cardinality.

5. **Define bijections between subsets:**
   Let \( \{g_i(n)\}_{i=1}^{k-1} \) be a set of bijections from \( A_i \) to \( A_{i+1} \). Also, define a bijection \( h(n) \) from \( A_k \) to \( A_1 \) as:
   \[
   h(n) = g_{1}^{-1}(g_{2}^{-1}(\cdots g_{k-1}^{-1}(n) \cdots))
   \]

6. **Define the function \( g(n) \):**
   Define \( g(n) \) from \( \mathbb{N} \to \mathbb{N} \) as follows:
   - If \( \frac{n}{k^{v_k(n)}} \in A_i \) with \( i \ne k \), then:
     \[
     g(n) = k^{v_k(n)} g_i \left( \frac{n}{k^{v_k(n)}} \right)
     \]
   - If \( \frac{n}{k^{v_k(n)}} \in A_k \), then:
     \[
     g(n) = k^{v_k(n) + 1} h \left( \frac{n}{k^{v_k(n)}} \right)
     \]

7. **Verify that \( g_k(n) = kn \):**
   By construction, the function \( g \) ensures that:
   \[
   g_k(n) = kn
   \]

8. **Define the function \( f(n) \):**
   Define \( f(n) \) as follows:
   - \( f(1) = 1 \)
   - For \( n > 1 \), let \( n = \prod p_i^{n_i} \) be the prime factorization of \( n \). Then:
     \[
     f(n) = \prod p_i^{g(n_i)}
     \]

9. **Verify the \( k \)-tastrophic property:**
   By the construction of \( f \), we ensure that:
   \[
   f_k(n) = n^k
   \]
   for all \( n > 0 \).

\(\blacksquare\)

The final answer is \( \boxed{ k > 1 } \).