Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:
[list]
[*] No two numbers are equal.
[*] If a number $m$ is placed on square $C$, then at least $k$ of the squares orthogonally adjacent to $C$ have a multiple of $m$ written on them.
[/list]
What is the greatest value of $k$ for which this is possible?
1. **Understanding the Problem:**
   We need to place positive integers on an infinite chessboard such that:
   - No two numbers are equal.
   - For any number \( m \) placed on a square \( C \), at least \( k \) of the squares orthogonally adjacent to \( C \) have a multiple of \( m \) written on them.

2. **Analyzing the Constraints:**
   - If \( k = 4 \), then all four orthogonally adjacent squares must have multiples of \( m \). This is impossible because two adjacent squares cannot both divide each other. Hence, \( k \neq 4 \).
   - If \( k = 3 \), consider a square with a number \( a \). Suppose the number \( ka \) is written next to it. Without loss of generality, let the number not divisible by \( a \) be to its left. Below \( ka \), there must be a multiple of it, say \( akb \). The square to the left of this must be a multiple of both \( a \) and \( akb \), which is not possible since it cannot divide both.

3. **Constructing a Solution for \( k = 2 \):**
   - We need to show that it is possible to place numbers such that each number \( m \) has at least 2 orthogonally adjacent squares with multiples of \( m \).
   - Start by placing the numbers \( 1, 2, 4, 8 \) in a \( 2 \times 2 \) square:
     \[
     \begin{array}{|c|c|}
     \hline
     1 & 2 \\
     \hline
     4 & 8 \\
     \hline
     \end{array}
     \]
   - We can extend this pattern by adding new rows or columns while maintaining the condition. For example, add a new column to the right:
     \[
     \begin{array}{|c|c|c|}
     \hline
     1 & 2 & 16 \\
     \hline
     4 & 8 & 32 \\
     \hline
     \end{array}
     \]
     Here, \( 16 \) is a multiple of \( 2 \) and \( 8 \), and \( 32 \) is a multiple of \( 4 \) and \( 8 \).

4. **Generalizing the Construction:**
   - For any new row or column, pick the top right square and add a multiple of it to its right. Below it, take a multiple of the product of numbers on the left and top (choose multiples large enough to ensure distinctness).
   - This ensures that each number has at least 2 orthogonally adjacent squares with multiples of it.

The final answer is \( \boxed{ k = 2 } \).