To solve the equation \( 27x^{3/4} - 1 = 26 \), we first add 1 to both sides:
\[ 27x^{3/4} = 27 \]
Next, we divide both sides by 27:
\[ x^{3/4} = 1 \]
To isolate \( x \), we raise both sides to the power of \( \frac{4}{3} \):
\[ x = 1^{\frac{4}{3}} = 1 \]
Thus, the solution to the equation is \( x = 1 \).
Therefore, the correct answer is:
\[ \boxed{1} \]