Summary: By iterating the Bolyai-Rényi transformation \(T(x)=(x+1)^{2}\pmod 1\), almost every real number \(x\in [0,1)\) can be expanded as a continued radical expression \[x=-1+\sqrt {x_{1}+\sqrt {x_{2}+\cdots +\sqrt{x_{n}+\cdots}}}\] with digits \(x_{n}\in\{0,1,2\}\) for all \(n\in\mathbb{N}\). For any real number \(x\in [0,1)\) and digit \(i\in\{0,1,2\}\), let \(r_{n}(x,i)\) be the maximal length of consecutive \(i\)’s in the first \(n\) digits of the Bolyai-Rényi expansion of \(x\). We study the asymptotic behavior of the run-length function \(r_{n}(x,i)\). We prove that for any digit \(i\in\{0,1,2\}\), the Lebesgue measure of the set \[D(i)=\Bigl\{x\in [0,1)\colon\lim_{n\rightarrow\infty}\frac{r_n(x,i)}{\log n}=\frac{1}{\log\theta_{i}}\Bigr\}\] is \(1\), where \(\theta_{i}=1+\sqrt{4i+1}\). We also obtain that the level set \[E_{\alpha}(i)=\Bigl\{x\in [0,1)\colon\lim_{n\rightarrow\infty}\frac{r_n(x,i)}{\log n}=\alpha\Bigr\}\] is of full Hausdorff dimension for any \(0\leq\alpha\leq\infty\).