Expected wealth when heads doubles and tails halves

You start with $1$. A fair coin is tossed repeatedly: on heads your wealth doubles and on tails it halves. What is your expected wealth after $n$ tosses, and what happens as $n$ grows large?

$(5/4)^n$, which explodes. Each toss multiplies wealth by $2$ or $\tfrac12$ with mean factor $\tfrac{2+0.5}{2}=1.25$. But $\log_2$ of wealth is a symmetric random walk, so the median stays near $1$: the growing mean comes from rare long runs of heads.