Write $2017$ following numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}$ .
One processes some steps as: erase two arbitrary numbers $x, y$ on the blackboard and then write on it the number $x + 7xy + y$. After $2016$ steps, there is only one number. The last one on the blackboard is

(A): $-\frac{1}{1008}$ (B): $0$ (C): $\frac{1}{1008}$ (D): $-\frac{144}{1008}$ (E): None of the above
1. **Initial Setup**: We start with the sequence of numbers on the blackboard:
   \[
   -\frac{1008}{1008}, -\frac{1007}{1008}, \ldots, -\frac{1}{1008}, 0, \frac{1}{1008}, \frac{2}{1008}, \ldots, \frac{1007}{1008}, \frac{1008}{1008}
   \]
   This sequence contains 2017 numbers.

2. **Operation Description**: The operation involves erasing two numbers \(x\) and \(y\) and writing the number \(x + 7xy + y\) on the blackboard.

3. **Invariant Identification**: We need to identify an invariant, a quantity that remains unchanged after each operation. Let's consider the sum of the reciprocals of the numbers on the blackboard.

4. **Invariant Calculation**: Initially, the sum of the numbers is:
   \[
   S = -\frac{1008}{1008} + -\frac{1007}{1008} + \cdots + -\frac{1}{1008} + 0 + \frac{1}{1008} + \frac{2}{1008} + \cdots + \frac{1007}{1008} + \frac{1008}{1008}
   \]
   Notice that the sequence is symmetric around 0, so the sum \(S\) is:
   \[
   S = 0
   \]

5. **Operation Analysis**: When we perform the operation \(x\) and \(y\) to get \(x + 7xy + y\), we need to check if the sum remains invariant:
   \[
   x + y \rightarrow x + 7xy + y
   \]
   The sum of the numbers before the operation is \(x + y\). After the operation, the new number is \(x + 7xy + y\). The sum of the numbers after the operation is:
   \[
   x + 7xy + y
   \]
   Since \(7xy\) is not zero in general, the sum of the numbers is not invariant. However, we need to check if there is a specific number that remains invariant.

6. **Special Number Identification**: Let's consider the number \(-\frac{1}{7}\). If we erase \(-\frac{1}{7}\) and another number \(y\), the new number is:
   \[
   -\frac{1}{7} + 7\left(-\frac{1}{7}\right)y + y = -\frac{1}{7} - y + y = -\frac{1}{7}
   \]
   This shows that \(-\frac{1}{7}\) remains invariant under the operation.

7. **Conclusion**: Since \(-\frac{1}{7}\) remains invariant and is present in the initial sequence (as \(-\frac{144}{1008} = -\frac{1}{7}\)), it will be the last number remaining after 2016 steps.

The final answer is \(\boxed{-\frac{1}{7}}\).