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