If this ends up being a ridiculous question I will delete it. Forgive me if this is ridiculous but this number has me stumped.
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.