Upward-biased walk ever reaching +1

A particle begins at $0$ and moves along the integers in steps of size one: each step is upward with probability $p=\tfrac23$ and downward otherwise, independently. What is the probability that the particle eventually visits position $1$?

$1$. Let $h$ be the probability of ever climbing one level. A first step up succeeds; a first step down requires climbing two levels, so $h=p+(1-p)h^2$, with roots $1$ and $p/(1-p)=2$. Since $h\le1$, $h=1$: a walk with upward drift reaches $+1$ almost surely.