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To solve the equation \( 27x^{3/4} - 1 = 26 \), we can start by isolating \( 27x^{3/4} \):
\[ 27x^{3/4} = 26 + 1 \]
\[ 27x^{3/4} = 27 \]
Next, we divide both sides by 27:
\[ x^{3/4} = 1 \]
Now, to solve for \( x \), we raise both sides to the power of \( \frac{4}{3} \):
\[ x = 1^{4/3} \]
\[ x = 1 \]
Therefore, the correct answer is \( \boxed{1} \).