According to Moor's Law, the number of shoes in Moor's room doubles every year.  In 2013, Moor's room starts out having exactly one pair of shoes.  If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatches pairs of shoes?  Note that left and right shoes are distinct, and Moor must always wear one of each.
1. According to Moor's Law, the number of pairs of shoes doubles every year. Starting in 2013 with 1 pair of shoes, the number of pairs of shoes in year \( n \) is given by:
   \[
   x_n = 2^{n-2013}
   \]
   where \( n \) is the year.

2. To find the earliest year when Moor can wear at least 500 mismatched pairs of shoes, we need to determine the number of mismatched pairs of shoes he can create with \( x \) pairs of shoes. For \( x \) pairs of shoes, the number of mismatched pairs is given by:
   \[
   \binom{x}{2} \cdot 2 = x(x-1)
   \]
   This is because for each pair of shoes, there are \( \binom{x}{2} \) ways to choose 2 pairs, and each pair can be mismatched in 2 ways (left shoe from one pair and right shoe from the other, and vice versa).

3. We need to find the smallest \( x \) such that:
   \[
   x(x-1) \geq 500
   \]

4. Solving the inequality:
   \[
   x^2 - x - 500 \geq 0
   \]
   Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -1 \), and \( c = -500 \):
   \[
   x = \frac{1 \pm \sqrt{1 + 2000}}{2} = \frac{1 \pm \sqrt{2001}}{2}
   \]
   Approximating \( \sqrt{2001} \approx 44.72 \):
   \[
   x \approx \frac{1 + 44.72}{2} \approx 22.86
   \]
   Since \( x \) must be an integer, we take \( x = 23 \).

5. Verify \( x = 23 \):
   \[
   23 \cdot 22 = 506 \geq 500
   \]
   So, \( x = 23 \) is sufficient.

6. Now, we need to find the year when Moor has 23 pairs of shoes. Since the number of pairs doubles every year:
   \[
   2^{n-2013} = 23
   \]
   Taking the logarithm base 2:
   \[
   n - 2013 = \log_2 23
   \]
   Approximating \( \log_2 23 \approx 4.52 \):
   \[
   n \approx 2013 + 4.52 \approx 2017.52
   \]
   Since \( n \) must be an integer, we take \( n = 2018 \).

Conclusion:
Moor will be able to wear at least 500 mismatched pairs of shoes in the year 2018.

The final answer is \( \boxed{2018} \).