Doubling-or-extinction population lifetime

A population of $10{,}000$ animals evolves year by year: each year, independently, it doubles with probability $0.9$ or is wiped out with probability $0.1$. What is the expected number of years until it goes extinct?

$10$. The population size is irrelevant; extinction happens in each year independently with probability $0.1$, so the waiting time is geometric with mean $1/0.1=10$. Equivalently $x=0.1\cdot1+0.9(1+x)$ gives $x=10$.