We know that $C_n-B_n=A_n^2$ for at least two different values of $n$.
We can rewrite this as $C_n-A_n^2=B_n$.
Since $C_n$ and $B_n$ are both positive integers, we must have $C_n>A_n^2$ for at least two different values of $n$.
This means that $C_n$ must be greater than the squares of all possible values of $A_n$.
Since $A_n$ is a nonzero digit, the largest possible value of $A_n$ is 9, so the largest possible value of $C_n$ is $9^2=81$.
Now, we need to find the largest possible value of $B_n$.
Since $B_n$ is a positive integer, it must be greater than or equal to 1.
Therefore, the largest possible value of $B_n$ is 1.
Finally, we can find the largest possible value of $a+b+c$ by considering the largest possible values of $A_n$, $B_n$, and $C_n$.
If $A_n=9$, $B_n=1$, and $C_n=81$, then $a=9$, $b=1$, and $c=8$, so $a+b+c=9+1+8=\boxed{18}$.
The answer is: 18
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
What is the largest possible value of $a + b + c$ that satisfies the condition $C_n - B_n = A_n^2$ for at least two different values of $n$, where $n$ is a positive integer and $a$, $b$, and $c$ are nonzero digits?

### Answer:
