A two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers.
1. **Identify the digits that are not prime:**
   - The digits that are not prime are \(1, 4, 6, 8, 9\).

2. **Identify the two-digit numbers ending in non-prime digits:**
   - Since we are looking for two-digit numbers that are prime but not primeable, we need to consider numbers ending in \(1\) or \(9\) (since these are the only non-prime odd digits).

3. **List the potential candidates:**
   - The two-digit numbers ending in \(1\) are: \(11, 21, 31, 41, 51, 61, 71, 81, 91\).
   - The two-digit numbers ending in \(9\) are: \(19, 29, 39, 49, 59, 69, 79, 89, 99\).

4. **Filter out the prime numbers from these candidates:**
   - From the list ending in \(1\): \(11, 31, 41, 61, 71\) are prime.
   - From the list ending in \(9\): \(19, 29, 59, 79, 89\) are prime.

5. **Check if these prime numbers are primeable:**
   - For \(11\): Deleting either digit gives \(1\), which is not prime. Hence, \(11\) is unripe.
   - For \(31\): Deleting \(3\) gives \(1\), which is not prime. Hence, \(31\) is unripe.
   - For \(41\): Deleting \(4\) gives \(1\), which is not prime. Hence, \(41\) is unripe.
   - For \(61\): Deleting \(6\) gives \(1\), which is not prime. Hence, \(61\) is unripe.
   - For \(71\): Deleting \(7\) gives \(1\), which is not prime. Hence, \(71\) is unripe.
   - For \(19\): Deleting \(1\) gives \(9\), which is not prime. Hence, \(19\) is unripe.
   - For \(29\): Deleting \(2\) gives \(9\), which is not prime. Hence, \(29\) is unripe.
   - For \(59\): Deleting \(5\) gives \(9\), which is not prime. Hence, \(59\) is unripe.
   - For \(79\): Deleting \(7\) gives \(9\), which is not prime. Hence, \(79\) is unripe.
   - For \(89\): Deleting \(8\) gives \(9\), which is not prime. Hence, \(89\) is unripe.

6. **Count the unripe integers:**
   - The unripe integers are \(11, 31, 41, 61, 71, 19, 29, 59, 79, 89\).

7. **Conclusion:**
   - There are \(10\) unripe integers.

The final answer is \(\boxed{10}\)