Answer: $\frac8{27}$


Using a diagram
 
first mouse eats $\frac13$ 
2 pieces left

second mouse eats $\frac13$ of the 2 pieces
split each piece into 3 - each piece $\frac19$
4 pieces left

third mouse eats $\frac13$ of the 4 pieces
split each piece into 3 - each piece $\frac1{27}$
8 pieces left
 

Direct calculation
After the first mouse has eaten, $\frac{2}{3}$ of the cheese remains.

After the second mouse has eaten, $\frac{2}{3}$ of $\frac{2}{3}$, that is $\frac{4}{9}$, of the cheese remains.

Finally, after the third mouse has eaten, $\frac{2}{3}$ of $\frac{4}{9}$, that is $\frac{8}{27}$ of the cheese remains.