There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
1. **Select a random turtle**: Without loss of generality (WLOG), we can consider a specific turtle and calculate the probability that this turtle is never chosen to be combined with another turtle.

2. **Calculate the probability for each step**: The probability that a specific turtle is not chosen in a single step when there are \( n \) turtles is given by:
   \[
   \frac{\binom{n-1}{2}}{\binom{n}{2}}
   \]
   This is because we need to choose 2 turtles out of the remaining \( n-1 \) turtles, and the total number of ways to choose 2 turtles out of \( n \) turtles is \(\binom{n}{2}\).

3. **Product of probabilities**: We need to calculate the product of these probabilities from \( n = 2017 \) down to \( n = 3 \):
   \[
   \frac{\binom{2016}{2}}{\binom{2017}{2}} \cdot \frac{\binom{2015}{2}}{\binom{2016}{2}} \cdot \frac{\binom{2014}{2}}{\binom{2015}{2}} \cdots \frac{\binom{3}{2}}{\binom{4}{2}} \cdot \frac{\binom{2}{2}}{\binom{3}{2}}
   \]

4. **Simplify the product**: Notice that each term in the product cancels out with the next term in the sequence:
   \[
   \frac{\binom{2016}{2}}{\binom{2017}{2}} \cdot \frac{\binom{2015}{2}}{\binom{2016}{2}} \cdot \frac{\binom{2014}{2}}{\binom{2015}{2}} \cdots \frac{\binom{3}{2}}{\binom{4}{2}} \cdot \frac{\binom{2}{2}}{\binom{3}{2}} = \frac{1}{\binom{2017}{2}}
   \]

5. **Calculate \(\binom{2017}{2}\)**: 
   \[
   \binom{2017}{2} = \frac{2017 \times 2016}{2} = 2017 \times 1008
   \]

6. **Probability for one turtle**: The probability that a specific turtle is never chosen is:
   \[
   \frac{1}{\binom{2017}{2}} = \frac{1}{2017 \times 1008}
   \]

7. **Total probability for any turtle**: Since there are 2017 turtles, the total probability that there is at least one turtle that has never been combined is:
   \[
   2017 \times \frac{1}{2017 \times 1008} = \frac{1}{1008}
   \]

8. **Simplify the fraction**: The fraction \(\frac{1}{1008}\) is already in its simplest form, so \( p = 1 \) and \( q = 1008 \).

9. **Sum of \( p \) and \( q \)**: 
   \[
   p + q = 1 + 1008 = 1009
   \]

The final answer is \( \boxed{1009} \)