Let's say we have continued fractions of irrational numbers of the form $$ [a_0, a_1, a_2,...]: a_0 \in \mathbb{Z}, a_i \in \{1,2\}. $$ Is there any way to determine a number, say $x\in [0,1],$ that cannot be written in the form described above?
EDIT: What about $x\in [0,1],$ that cannot be written as a sum x=a+b, where $a,b$ are of the above form?
You can certainly generate irrational numbers in $[0,1]$ that don't have continued fractions satisfying your criterion. Note that irrational numbers have unique continued fractions. A continued fraction $[a_0;a_1,a_2,...]$ with $a_0=0$ lies in $[0,1)$ irrespective of subsequent partial denominators. Thus, any simple infinite continued fraction of the form $[0;a_1,a_2,...]$ where one or more $a_i>2,i>0$ uniquely represents an irrational in $[0,1)$.