Fibonacci used a different version of continued fraction which I'm curious about. It seems the notation is equivalent to $$[a_1,a_2,a_3,\cdots]=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\cdots,$$ where $a_k$ are positive integers and (I believe) it is required that $1 \le a_1 \le a_2 \le a_3 \le \cdots .$ The reason for my belief is that if we drop the nondecreasing assumption uniqueness is lost.
What I'd like to see is a proof (or reference to one) that each positive real has a unique such expression. Or just a reference to this type of "continued fraction" which I could follow up on.
Let $x_1$ be a positive real number. Take $a_1>0$ minimal such that $\frac{1}{a_1} < x_1$ and set $x_2 = a_1x_1 - 1$. Note that $x_2$ is positive and by the minimality of $a_1$ we have $$ x_1 \le \frac{1}{a_1-1} $$ which is equivalent to $x_2\le x_1$. Now letting $a_2>0$ be minimal such that $\frac{1}{a_2} < x_2$ we must have $a_2\ge a_1$ and obtained $$ x_1 = \frac{1}{a_1} + \frac{x_2}{a_1} = \frac{1}{a_1} + \frac{1}{a_1a_2} + \frac{x_2a_2-1}{a_1a_2}. $$ Repeating this, setting $x_{i+1} = x_i a_i-1$ in each step, should yield the desired unique expansion.