I have found a number that satisfies this continued fraction:$$n=1+{1\over{2+{1\over{3+{1\over n}}}}}$$ With a value of about $1.4403$ after 9 layers of nesting. I've tried googling it and plugging it into Wolfram Alpha, but it doesn't seem to be used. Does it have any application or is some root of another number?
ADDENDUM
It has the exact value of $${13\pm(\sqrt{37}+\frac{1}{7}})\over{7}$$
The OEIS sequence A060997 is the decimal expansion of the continued fraction $\,1+1/(2+1/3(1+1/4+\dots)))\,$ with value $1.433127426722311758317183455775\dots$ and the exact formula is $\,I_0(2)/I_1(2)\,$ where $\,I_n(x)\,$ is the modified Bessel function of the first kind.
For the continued fraction $\,1+1/(2+1/(3+1/(1+1/(2+1/3+\dots))))) ,\,$ solving the equation $\, x =1+1/(2+1/(3+1/x)) \,$ simplifies to the quadratic equation $\, 0 = 7x^2 - 8x - 3 \,$ with solution $\, x = (4 + \sqrt{37})/7 \approx 1.440394647185.\,$ The decimal expansion is the sequence A177036 whose continued fraction is sequence A010882.