Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau_x$ be the time to reach $x$ and condition on the event $\tau_M < \tau_0$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V_k$ be the number of visits to $k$ up to time $\tau_M$. What is $\mathbb E V_k$?

All we need is a bound independent of $M$. Something like there exists $C>0$ such that

$$\sup_{M >0 }\Bigl[ \max_{0< k < M} \mathbb E V_k \Bigr]< C.$$