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 $\boxed{-\frac{1}{7}}$.