Preprocessing Table

Create a table where $\text{jump}[i][x]$ stores $f^{2^i}(x)$. Base case: $\text{jump}[0][x] = f(x)$. This is only one step. For $i \geq 1$: $\text{jump}[i][x] = \text{jump}[i-1][\text{jump}[i-1][x]]$. Jumping $2^i$ steps is jumping $2^{i-1}$ steps twice.

Build the table bottom-up. Store up to $i = \lfloor \log_2 n \rfloor$ since you cannot jump more than $n$ steps without repeating in a cycle. This bounds the table size.