Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.)

What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)
1. **Define the problem and notation:**
   Given a positive integer \( N \) written in base 10, we need to find the greatest integer \( N \) such that no integer substring of \( N \) is a multiple of 9. 

2. **Define integer substrings:**
   Integer substrings of \( N \) are integers formed by consecutive digits of \( N \). For example, for \( N = 3208 \), the integer substrings are \( 3, 2, 0, 8, 32, 20, 320, 208, 3208 \).

3. **Use properties of divisibility by 9:**
   A number is divisible by 9 if the sum of its digits is divisible by 9. Let \( N = \overline{d_1d_2 \dots d_n} \) where \( d_i \) are the digits of \( N \). Define \( p(k) = \sum_{\ell = 1}^k d_\ell \) for each \( 1 \le k \le n \) and let \( p(0) = 0 \).

4. **Express integer substrings in terms of \( p(k) \):**
   The integer substring \( \overline{d_i d_{i+1} \dots d_j} \) can be expressed as:
   \[
   \overline{d_i d_{i+1} \dots d_j} \equiv \sum_{\ell = i}^j d_\ell = p(j) - p(i-1) \pmod{9}
   \]
   Hence, \( 9 \mid \overline{d_i d_{i+1} \dots d_j} \) if and only if \( p(j) \equiv p(i-1) \pmod{9} \).

5. **Ensure distinctness modulo 9:**
   To ensure no integer substring of \( N \) is a multiple of 9, \( p(0), p(1), \ldots, p(n) \) must be distinct modulo 9.

6. **Count the number of distinct values modulo 9:**
   Since there are 9 possible remainders when divided by 9 (0 through 8), if \( n \ge 9 \), then \( p(0), p(1), \ldots, p(n) \) will have at least 10 elements, meaning at least two elements must be congruent modulo 9 by the pigeonhole principle. Therefore, \( n \le 8 \).

7. **Check the maximum value of \( N \):**
   If \( n = 8 \), then \( N \) can have at most 8 digits. Since no digit can be 9 (as it would be a multiple of 9), the largest possible value for \( N \) is \( 88888888 \).

8. **Verify the solution:**
   For \( N = 88888888 \), we need to check that no integer substring is a multiple of 9. Since each digit is 8, the sum of any substring's digits will be a multiple of 8, not 9. Thus, \( 88888888 \) satisfies the condition.

The final answer is \( \boxed{88888888} \).