I was wondering if there is some positive real $x<1$ with non-terminating decimal expansion $b_1b_2b_3\dots$ where $b_1\neq0$ which also satisfies $$x=\dfrac1{b_1b_2\dots b_i+\dfrac1{b_{i+1}b_{i+2}\dots b_{j} + \dfrac1{b_{j+1}\dots b_k + \dfrac1{\ddots}}}}$$ Where $b_gb_{g+1}\dots$ is not a product, rather some number with decimal expansion $b_gb_{g+1}\dots$. For example, if $b_1=2$ and $b_2=4$, then $b_1b_2$ is the number $24$, not $8$.
I have found a family of rational $x$ which satisfy the above. They are: $$\frac13 = \frac1{3+\dfrac1{333333\dots + \dfrac1{\text{doesn't matter what goes here.}}}}$$ $$\frac1{33} = \frac1{3+\dfrac1{03 + \dfrac1{03+\dfrac1{03030303030303\dots + \dfrac1{\text{doesn't matter what goes here.}}}}}}$$ And so forth.
My question: can we prove/disprove the existence of an irrational such $x$? Can we determine the value of such an $x$, if it exists? Also, does my family of rational $x$ being $\frac13,\frac1{33}\dots$ cover all rational solutions?