Let the i.i.d. binary variables $B_{1}, B_{2}, \ldots$ with $P(B_{k}=1)=\beta$ and $P(B_{k}=0) = 1-\beta$, with $\beta \in (0,1)$. Define the variable $M$ as follows: $M = \max\{k\geq 0: \prod_{i=0}^{k}B_{i} = 1\}$.
The problem is to find the expected value of $M$, $\mathbb{E}[M]$.
Given feedback, this reflects to the variable $M$ following the geometric distribution, where $M$ is the largest integer before the first success, thus, $E[M] = \frac{1}{1-\beta}$.
I presume you mean $M=\max\{k\ge 0: \prod_{i=1}^k B_i=1\}$, i.e. the product is taken from $1$, not zero, with the convention that if $B_1=0$ then $M=0$. You can compute the pmf of $M$ directly: \begin{align} \mathbb{P}(M=m) &= \mathbb{P}(B_1=B_2=\dots=B_m=1, B_{m+1}=0) \\ &=\mathbb{P}(B_1=1)\mathbb{P}(B_2=1)\cdots\mathbb{P}(B_m=1)\mathbb{P} (B_{m+1}=0) \\ &=\beta^m(1-\beta). \end{align} Now $$\mathbb{E}M=\sum_{m=0}^\infty m\mathbb{P}(M=m)=\sum_{m=0}^\infty m\beta^m(1-\beta)=\beta(1-\beta)\sum_{m=0}^\infty m\beta^{m-1}=\beta(1-\beta)\cdot \frac1{(1-\beta)^2}.$$