C

Step 1: Identify the Configuration
The given circuit consists of resistors arranged in a combination of series and parallel. Starting from point A, we have:
- A resistor of value 2R in series with a combination of resistors below it (R, 9R, and 6R).

Step 2: Solve the Parallel Combination
The three resistors (R, 9R, and 6R) are in parallel. The formula for the total resistance (R_parallel) for resistors in parallel is given by:
$$ \frac{1}{R_{parallel}} = \frac{1}{R} + \frac{1}{9R} + \frac{1}{6R} $$
Finding a common denominator (18R), we get:
$$ \frac{1}{R_{parallel}} = \frac{18}{18R} + \frac{2}{18R} + \frac{3}{18R} = \frac{23}{18R} $$
Thus,
$$ R_{parallel} = \frac{18R}{23} $$

Step 3: Total Resistance Calculation
Now, this equivalent resistance (R_parallel) is in series with the 2R resistor at the top. Therefore, the total resistance (R_total) between A and B can be calculated as follows:
$$ R_{total} = 2R + R_{parallel} = 2R + \frac{18R}{23} $$
To add these together, express 2R with a common denominator:
$$ R_{total} = \frac{46R}{23} + \frac{18R}{23} = \frac{64R}{23} $$

Final Answer
The equivalent resistance between A and B is therefore:
$$ R_{eq} = \frac{64R}{23} $$ which corresponds to option C.