The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases}  \] Find $f(84)$.
1. We start with the given function definition:
   \[
   f(n) = \begin{cases} 
   n - 3 & \text{if } n \ge 1000 \\
   f(f(n + 5)) & \text{if } n < 1000 
   \end{cases}
   \]
   We need to find \( f(84) \).

2. Since \( 84 < 1000 \), we use the second case of the function definition:
   \[
   f(84) = f(f(89))
   \]

3. Next, we need to find \( f(89) \). Since \( 89 < 1000 \), we again use the second case:
   \[
   f(89) = f(f(94))
   \]

4. Continuing this process, we need to find \( f(94) \). Since \( 94 < 1000 \), we use the second case:
   \[
   f(94) = f(f(99))
   \]

5. Now, we need to find \( f(99) \). Since \( 99 < 1000 \), we use the second case:
   \[
   f(99) = f(f(104))
   \]

6. Next, we need to find \( f(104) \). Since \( 104 \ge 1000 \), we use the first case:
   \[
   f(104) = 104 - 3 = 101
   \]

7. Now, we substitute back to find \( f(99) \):
   \[
   f(99) = f(101)
   \]

8. Since \( 101 \ge 1000 \), we use the first case:
   \[
   f(101) = 101 - 3 = 98
   \]

9. Now, we substitute back to find \( f(94) \):
   \[
   f(94) = f(98)
   \]

10. Since \( 98 < 1000 \), we use the second case:
    \[
    f(98) = f(f(103))
    \]

11. Next, we need to find \( f(103) \). Since \( 103 \ge 1000 \), we use the first case:
    \[
    f(103) = 103 - 3 = 100
    \]

12. Now, we substitute back to find \( f(98) \):
    \[
    f(98) = f(100)
    \]

13. Since \( 100 \ge 1000 \), we use the first case:
    \[
    f(100) = 100 - 3 = 97
    \]

14. Now, we substitute back to find \( f(94) \):
    \[
    f(94) = f(97)
    \]

15. Since \( 97 < 1000 \), we use the second case:
    \[
    f(97) = f(f(102))
    \]

16. Next, we need to find \( f(102) \). Since \( 102 \ge 1000 \), we use the first case:
    \[
    f(102) = 102 - 3 = 99
    \]

17. Now, we substitute back to find \( f(97) \):
    \[
    f(97) = f(99)
    \]

18. We already found \( f(99) = 98 \), so:
    \[
    f(97) = 98
    \]

19. Now, we substitute back to find \( f(94) \):
    \[
    f(94) = 98
    \]

20. Now, we substitute back to find \( f(89) \):
    \[
    f(89) = f(98)
    \]

21. We already found \( f(98) = 97 \), so:
    \[
    f(89) = 97
    \]

22. Now, we substitute back to find \( f(84) \):
    \[
    f(84) = f(97)
    \]

23. We already found \( f(97) = 98 \), so:
    \[
    f(84) = 98
    \]

The final answer is \( \boxed{98} \).