If this ends up being a ridiculous question I will delete it. Forgive me if this is ridiculous but this number has me stumped.
$$1.52360679774998$$
The continued fraction calculator gives $1, 1, 1, 10, 11, 11, 11, 11, 11, 11 ...$ which makes me think this number should have a nice expression as a root or log of something or be related to some special number like $\phi$. But I've been unable to tease out any such expression. I appreciate it if someone has more insight into this.
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Answer: $$\frac{13+\sqrt{5}}{10} \approx 1.52360679774997896964091736687...$$
(I got the result using the Inverse Symbolic Calculator: http://isc.carma.newcastle.edu.au )
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As I had said in the comments, Wolfram Alpha gives multiple closed form solutions for numbers. The first three are:
$$ \frac{1}{10}\left(13+\sqrt{5}\right), \quad \frac{\Phi+7}{5}, \quad \frac{1}{5\Phi}+\frac{6}{5} $$
where $\Phi$ is the golden ratio conjugate.
If we assume that the continued fraction expansion that you quoted continues in that pattern forever, we can do the calculation by hand. For let $x$ be the value of the continued fraction $\langle 0;11,11,11,\dots\rangle$. Then $x=\dfrac{1}{11+x}$. This gives a quadratic equation with positive root $\dfrac{5\sqrt{5}-9}{2}$.
Now we can claw our way to the top. For example, $\langle 0;10,11,11,11,\dots\rangle=\dfrac{1}{10+x}=\dfrac{2}{5\sqrt{5}+9}$. Continue, resisting the urge to rationalize the denominator. At the end we get $\dfrac{5\sqrt{5}+31}{10\sqrt{5}+20}$. Finally, rationalize the denominator. We get $\dfrac{13+\sqrt{5}}{10}$.