On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$. Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.
1. **Initial Sum Calculation:**
   The sum of the numbers from 1 to 9 is:
   \[
   1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
   \]

2. **Effect of Adding and Subtracting Signs:**
   If we place a minus sign between any two numbers, it effectively subtracts twice the value of the number following the minus sign from the total sum. For example, if we place a minus sign before the number \(a\), the contribution of \(a\) to the total sum changes from \(+a\) to \(-a\), which is equivalent to subtracting \(2a\) from the total sum.

3. **General Formulation:**
   Let \(S\) be the sum of the numbers from 1 to 9, which is 45. If we place minus signs before some of the numbers, the resulting sum can be written as:
   \[
   45 - 2 \times (\text{sum of the numbers with minus signs})
   \]
   Let \(T\) be the sum of the numbers with minus signs. Then the resulting sum is:
   \[
   45 - 2T
   \]

4. **Range of \(T\):**
   The possible values of \(T\) are the sums of any subset of the numbers from 2 to 9. We need to determine the range of possible values for \(T\):
   - The smallest possible value of \(T\) is 2 (if only 2 is chosen).
   - The largest possible value of \(T\) is \(2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 44\).

5. **Possible Values of \(45 - 2T\):**
   We need to find the possible values of \(45 - 2T\) for \(T\) ranging from 2 to 44. The expression \(45 - 2T\) will yield odd values when \(T\) is an integer.

6. **Finding the Least Odd Positive Integer That Cannot Be Obtained:**
   We need to check which odd positive integers cannot be obtained by \(45 - 2T\). We start by listing the possible values of \(45 - 2T\) for \(T\) from 2 to 44:
   - For \(T = 2\), \(45 - 2 \times 2 = 45 - 4 = 41\)
   - For \(T = 3\), \(45 - 2 \times 3 = 45 - 6 = 39\)
   - For \(T = 4\), \(45 - 2 \times 4 = 45 - 8 = 37\)
   - Continue this process until \(T = 22\), \(45 - 2 \times 22 = 45 - 44 = 1\)

   The possible values of \(45 - 2T\) are: 41, 39, 37, 35, 33, 31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1.

   We observe that all odd numbers from 1 to 41 are covered except for 43.

Conclusion:
The least odd positive integer that Andi cannot obtain is 43.

The final answer is \(\boxed{43}\)