We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$.  Compute the greatest possible weight of a positive integer less than 100.
To solve the problem, we need to compute the weight \( W(n) \) for positive integers \( n \) and determine the greatest possible weight for \( n < 100 \). The weight function \( W \) is defined recursively as follows:
- \( W(1) = 0 \)
- \( W(2) = 1 \)
- \( W(p) = 1 + W(p + 1) \) for every odd prime \( p \)
- \( W(c) = 1 + W(d) \) for every composite \( c \), where \( d \) is the greatest proper factor of \( c \)

Let's compute the weights for the first few integers and then use these values to find the greatest possible weight for \( n < 100 \).

1. **Compute the weights for the first 13 integers:**
   \[
   \begin{array}{r|rrrrrrrrrrrrr}
   n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\
   \hline
   W(n) & 0 & 1 & 3 & 2 & 5 & 4 & 4 & 3 & 4 & 6 & 6 & 5 & 6
   \end{array}
   \]

2. **Verify the weights for \( n \leq 13 \):**
   - \( W(1) = 0 \)
   - \( W(2) = 1 \)
   - \( W(3) = 1 + W(4) = 1 + 2 = 3 \)
   - \( W(4) = 1 + W(2) = 1 + 1 = 2 \)
   - \( W(5) = 1 + W(6) = 1 + 4 = 5 \)
   - \( W(6) = 1 + W(3) = 1 + 3 = 4 \)
   - \( W(7) = 1 + W(8) = 1 + 3 = 4 \)
   - \( W(8) = 1 + W(4) = 1 + 2 = 3 \)
   - \( W(9) = 1 + W(3) = 1 + 3 = 4 \)
   - \( W(10) = 1 + W(5) = 1 + 5 = 6 \)
   - \( W(11) = 1 + W(12) = 1 + 5 = 6 \)
   - \( W(12) = 1 + W(6) = 1 + 4 = 5 \)
   - \( W(13) = 1 + W(14) = 1 + 5 = 6 \)

3. **Extend the computation to \( n \leq 26 \):**
   - If \( n \leq 13 \), then \( W(n) \leq 6 \).
   - If \( n \) is composite, its greatest proper factor \( d \leq \frac{n}{2} \leq 13 \), so \( W(n) = 1 + W(d) \leq 1 + 6 = 7 \).
   - If \( n \) is an odd prime, \( W(n) = 1 + W(n + 1) = 1 + 1 + W(\frac{n + 1}{2}) \leq 1 + 1 + 6 = 8 \).

4. **Extend the computation to \( n \leq 52 \):**
   - If \( n \leq 26 \), then \( W(n) \leq 8 \).
   - If \( n \) is composite, its greatest proper factor \( d \leq \frac{n}{2} \leq 26 \), so \( W(n) = 1 + W(d) \leq 1 + 8 = 9 \).
   - If \( n \) is an odd prime, \( W(n) = 1 + W(n + 1) = 1 + 1 + W(\frac{n + 1}{2}) \leq 1 + 1 + 8 = 10 \).

5. **Extend the computation to \( n \leq 104 \):**
   - If \( n \leq 52 \), then \( W(n) \leq 10 \).
   - If \( n \) is composite, its greatest proper factor \( d \leq \frac{n}{2} \leq 52 \), so \( W(n) = 1 + W(d) \leq 1 + 10 = 11 \).
   - If \( n \) is an odd prime, \( W(n) = 1 + W(n + 1) = 1 + 1 + W(\frac{n + 1}{2}) \leq 1 + 1 + 10 = 12 \).

6. **Find a positive integer less than 100 with weight 12:**
   - Consider the sequence of primes: \( 19 \), \( 37 \), and \( 73 \).
   - \( W(73) = 1 + W(74) = 1 + 1 + W(37) = 2 + W(37) \)
   - \( W(37) = 1 + W(38) = 1 + 1 + W(19) = 2 + W(19) \)
   - \( W(19) = 1 + W(20) = 1 + 1 + W(10) = 2 + W(10) \)
   - \( W(10) = 6 \)
   - Therefore, \( W(19) = 2 + 6 = 8 \)
   - \( W(37) = 2 + 8 = 10 \)
   - \( W(73) = 2 + 10 = 12 \)

Thus, the greatest possible weight of a positive integer less than 100 is \( \boxed{12} \).