TITLE: On the "Look-and-Say" sequence and Conway's constant
QUESTION [17 upvotes]: The look-and-say sequence starting with $S_1=1$ is,
$$S_n = 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,\dots$$
If $L_n$ is the number of digits of the $n$th term then,
$$\lim_{n\to\infty} \frac{L_{n+1}}{L_n}=\lambda\tag{1}$$
where $\lambda = 1.303577\dots$ is an algebraic number of degree 71. 
(A discussion of this sequence and the constant can be found in Nathaniel Johnston's blog.)
Questions:

Is the limit (1) an artifact of base-10? If the terms of $S_n$ are translated into some other base-$n$ representation, say in binary, will the limit still be $\lambda$?
I find it intriguing that $\lambda$ happens to be an algebraic number of such high degree. Without using obvious examples like $x^n=1$, or n-nacci constants $x^n(2-x)=1$, or contrived ones $\phi^{1/m}, \lambda^{1/n}$ etc, is there a constant that appears in a non-trivial number-theoretic context (like a limiting ratio, etc) that is algebraic and has a higher degree than $\lambda$?

REPLY [3 votes]: 1) You can use a similar approach to Nathaniel Johnston's Blog for the binary look-and-say sequence:
1,11,101,111011,11110101,...
You can do it differently I suppose, but I got those 10 subsequences:
111011,11110101,100110,11100,10110,1110,111100,1001100,11110,101100.
You get a transformationmatrix with charactaristical polynomial
$$
x^4(x+1)(x-1)^2(x^3-x^2-1)
$$
Which has a non-trivial positive real root
$$
x\approx 1.465571232
$$