Is anything known about $ \small{b_0+\tfrac{a_1}{\left(b_1+\tfrac{a_2}{\left(b_2+...\right)^n}\right)^n}} $? What is known about this generalized "continued fraction"
$$
b_0+\frac{a_1}{\left(b_1+\frac{a_2}{\left(b_2+\frac{a_3}{\left(b_3+\dotsb\right)^n}\right)^n}\right)^n}
$$
when the integer $n\ge 2$?
Wikipedia and wolfram articles on generalized continued fraction doesn't mention any continued fractions of this kind.
Of course one can calculate explicitly periodic continued fractions, but they are not interesting.

Not interesting! Explicitly periodic generalized continued fractions are not interesting?  You do realize that if you take $n=2$, and you assume a period of $k$, then your continued fraction (if it converges) is the root of a degree $2k+1$ polynomial?
Explicitly: If $x = [a,a,a,a,...;b,b,b,b,...]_2$, then $x = a + b / x^2$, and $x^3 - ax^2 - b = 0$.  If $x = [a, a', a, a',...; b,b',b,b',...]_2$, then $x = a + b / (a' + b'/x^2)^2 = a + bx^4 / (a'x^2 + b')^2$ or $a'^2 x^5 - (aa'^2 + b) x^4 + 2a'b' x^3 - 2aa'b' x^2 + b'^2 x - ab'^2 = 0$. Setting $c = b/a'^2$ and $d = (b'/a')$, we get $x^5 - (a+c)x^4 + 2dx^3 - 2adx^2 +d^2x - ad^2 = 0$.
What I find interesting is that in the latter case, you don't have the full space of quintic polynomials -- you only get solutions to a 3-dimensional subspace.
