I decided to compute the ratio $x_{30}/x_{29}$ for various start values $x_0 = x_1 = s$ For $s>0.42$, the computations overflows for me, so I could not compute that part.
The image shows the ratio on the y axis, and start value on the x axis. The images are essentially identical for $x_{31}/x_{30}$, so it is motivated to take 30 as an approximation of $\infty.$
EDIT: So here is plots for $x_0 = s$ for different values of $s$.
http://www2.math.su.se/~per/files.php?file=recursiondata_mathoverflow_87463.pdf
