To solve the expression \((1/6)^{2.6^2} + \frac{(0.6)^5}{(0.2)^6}\), we need to break it down into parts and solve each part step by step.

### Part 1: \((1/6)^{2.6^2}\)

1. Calculate \(2.6^2\):
\[
2.6^2 = 2.6 \times 2.6 = 6.76
\]

2. Raise \(\frac{1}{6}\) to the power of 6.76:
\[
\left(\frac{1}{6}\right)^{6.76}
\]

### Part 2: \(\frac{(0.6)^5}{(0.2)^6}\)

1. Calculate \((0.6)^5\):
\[
(0.6)^5 = 0.6 \times 0.6 \times 0.6 \times 0.6 \times 0.6 = 0.07776
\]

2. Calculate \((0.2)^6\):
\[
(0.2)^6 = 0.2 \times 0.2 \times 0.2 \times 0.2 \times 0.2 \times 0.2 = 0.000064
\]

3. Divide the results:
\[
\frac{0.07776}{0.000064} = 1215
\]

### Combine the Results

Now, we combine the results from both parts:
\[
(1/6)^{6.76} + 1215
\]

Since \((1/6)^{6.76}\) is a very small number, it can be approximated to a value close to zero. Therefore, the dominant term in the expression is 1215.

Thus, the approximate value of the expression is:
\[
1215
\]