Time to climb one step with downward drift

A particle starts at $0$ on the integer line. Each step it moves up by one with probability $\tfrac13$ and down by one with probability $\tfrac23$, independently. What is the expected number of steps until it first visits $1$? (State whether the expectation is infinite.)

Infinite. With downward drift, the walk reaches $+1$ only with probability $\tfrac{p}{1-p}=\tfrac12<1$. Since there is positive probability of never arriving, the hitting time is infinite with positive probability and so its expectation is infinite.