Say we have a simple continued fraction with rapidly increasing terms. Then it obviously converges very quickly and has a very good rational approximation.
But is there anything special or interesting about such numbers? They are obviously irrational and most likely transcendental, but maybe they have some other curious properties?
For example, consider the factorial continued fraction:
$$ b_1=\dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{6+}}} $$ $$ b_1=[0;1!,2!,3!,4!,5!,\dots] \approx 0.684095900106622500 $$
or this one:
$$ b_2=\dfrac{1}{1+\dfrac{1}{4+\dfrac{1}{27+}}} $$ $$ b_2=[0;1^1,2^2,3^3,4^4,5^5,\dots] \approx 0.801470377071571461 $$
Anyway, is there a comprehensive study of various regular continued fractions like there is for infinite series?