In my posts, I always use the following notation for the continued fractions:

$$ a_0 + \operatorname*{K}_{n=1}^{\infty} \frac{b_n}{a_n} = a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + \dotsb}}} $$

(see for instance here, here or here).

I find it convenient and it can be found for instance in Continued Fractions with Applications by Lorentzen & Waadeland.

The first place where I saw it was on Wikipedia where it is said:

Carl Friedrich Gauss evoked the more familiar infinite product Π when he devised this notation: $$ {\displaystyle x=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}.\,} $$ Here the "K" stands for Kettenbruch, the German word for "continued fraction". This is probably the most compact and convenient way to express continued fractions; however, it is not widely used by English typesetters.

I browsed (very quickly) all articles from Gauss where the topic of continued fractions was obvious without seeing it once; for this reason I am not sure the statement on this page is true.

Would someone have a more precise reference?