Does this number have an expression as square root or log of something 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.

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 $\dfrac{13+\sqrt{5}}{10}$.
