How many $3$ s are there in the following sequence which are neither preceded by $6$ nor immediately followed by $9$?
$$\text{9 3 6 6 3 9 5 9 3 7 8 9 1 6 3 9 6 3 9}$$

We need to find $3$ s such that:
* The digit before $3$ is NOT $6$.
* The digit after $3$ is NOT $9$.

Let us examine each $3$ in the sequence:

1. $9-3-6$: Preceded by $9$ (Not $6$), followed by $6$ (Not $9$). $\to$ Valid
2. $6-3-9$: Preceded by $6$. $\to$ Invalid
3. $9-3-7$: Preceded by $9$ (Not $6$), followed by $7$ (Not $9$). $\to$ Valid
4. $6-3-9$: Preceded by $6$. $\to$ Invalid
5. $6-3-9$: Preceded by $6$. $\to$ Invalid

There are exactly $2$ such $3$ s that satisfy the given conditions.