To find the experimental probability that the band will play an encore at its next show, we use the formula for experimental probability:
\[ P(\text{encore}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of trials}} \]
In this case, the number of favorable outcomes (the shows where they played an encore) is 2, and the total number of trials (the total shows) is 6.
So we calculate:
\[ P(\text{encore}) = \frac{2}{6} \]
Now we can simplify the fraction:
\[ P(\text{encore}) = \frac{1}{3} \]
Therefore, the experimental probability that the band will play an encore at its next show is:
\[ \boxed{\frac{1}{3}} \]