Nine numbers $a, b, c, \dots$ are arranged around a circle. All numbers of the form $a+b^c, \dots$ are prime. What is the largest possible number of different numbers among $a, b, c, \dots$?
1. **Understanding the Problem:**
   We are given nine numbers \(a, b, c, \dots\) arranged in a circle such that all expressions of the form \(a + b^c\) are prime. We need to determine the largest possible number of different numbers among \(a, b, c, \dots\).

2. **Initial Assumptions:**
   Let's assume the numbers are natural numbers. This is a reasonable assumption since the problem involves prime numbers, which are natural numbers greater than 1.

3. **Example and Pattern:**
   Consider the example given: \(1, 2, 1, 4, 1, 6, 1, 10, 1\). This example works because:
   - \(1 + 2^1 = 3\) (prime)
   - \(2 + 1^4 = 3\) (prime)
   - \(1 + 4^1 = 5\) (prime)
   - \(4 + 1^6 = 5\) (prime)
   - \(1 + 6^1 = 7\) (prime)
   - \(6 + 1^{10} = 7\) (prime)
   - \(1 + 10^1 = 11\) (prime)
   - \(10 + 1^2 = 11\) (prime)

4. **Analysis of Even and Odd Numbers:**
   - Note that there are no two even numbers in a row. This is because if there were two consecutive even numbers, their sum would be even, and the only even prime number is 2. However, \(a + b^c\) would not be 2 in general.
   - Since there are 9 numbers (an odd count), there must be at least two consecutive odd numbers. These two odd numbers must both be 1 to satisfy the prime condition.

5. **Limiting the Number of Different Numbers:**
   - If there are more than 4 numbers greater than 1, there would be two adjacent numbers both greater than 1. Let's denote these numbers as \(a\) and \(b\) with \(a, b > 1\).
   - Consider the sequence \(1, a, b, c\) with \(a, b > 1\). For \(1 + a^b\) to be prime, \(a\) must be even (since \(a^b\) is odd if \(a\) is even and \(b\) is odd).
   - For \(a + b^c\) to be prime, \(b\) must be odd (since \(b^c\) is odd if \(b\) is odd and \(c\) is even).

6. **Contradiction:**
   - If \(a\) is even and \(b\) is odd, then \(a + 1\) divides \(1 + a^b\) and \(a + 1 < 1 + a^b\), which means \(1 + a^b\) cannot be prime. This is a contradiction.

7. **Conclusion:**
   - Therefore, the maximum number of different numbers among \(a, b, c, \dots\) is 5, as shown in the example.

The final answer is \(\boxed{5}\).