Calculating the Value of the Expression
To find the value of the given expression, we need to simplify each term step by step using the rules of exponents.

Simplifying the Terms
1. \( \frac{4}{{216}^{-\frac{2}{3}}} \)
Using the property of negative exponents, we can rewrite \( {216}^{-\frac{2}{3}} \) as \( \frac{1}{{216}^{\frac{2}{3}}} \).
Now, \( 216 = 6^3 \), so \( {216}^{\frac{2}{3}} = (6^3)^{\frac{2}{3}} = 6^2 = 36 \).
Therefore, \( \frac{4}{36} = \frac{1}{9} \).
2. \( \frac{1}{{256}^{-\frac{3}{4}}} \)
Similarly, using the property of negative exponents, we rewrite \( {256}^{-\frac{3}{4}} \) as \( \frac{1}{{256}^{\frac{3}{4}}} \).
Now, \( 256 = 4^4 \), so \( {256}^{\frac{3}{4}} = (4^4)^{\frac{3}{4}} = 4^3 = 64 \).
Therefore, \( \frac{1}{64} \).
3. \( \frac{2}{{243}^{-\frac{1}{5}}} \)
Again, using the property of negative exponents, we rewrite \( {243}^{-\frac{1}{5}} \) as \( \frac{1}{{243}^{\frac{1}{5}}} \).
Now, \( 243 = 3^5 \), so \( {243}^{\frac{1}{5}} = (3^5)^{\frac{1}{5}} = 3 \).
Therefore, \( 2 \times 3 = 6 \).

Calculating the Final Value
Adding up the simplified terms, we get:
\( \frac{1}{9} + \frac{1}{64} + 6 = \frac{64 + 9 + 576}{576} = \frac{649}{576} \).
Therefore, the value of the expression is \( \frac{649}{576} \).