"base" expansions vs. continued fractions (Are any the same?)

Question

Does anyone know of any examples of irrational numbers for which their continued fractions "equal" their expansion in some base-number system? More concretely, does there exists some $x\in\mathbb{R}\setminus\mathbb{Q}$, and some $p\in\mathbb{Z}^+$, for which $$ x=[a_0;a_1,a_2,a_3,\ldots] = \frac{a_0}{p^0}+\frac{a_1}{p^1}+\frac{a_2}{p^2}+\frac{a_3}{p^3}+\cdots, $$ where $[a_0;a_1,a_2,a_3,\ldots]$ means $$ a_0 + \frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\ddots}}} $$