Waiting for a uniform to beat a random threshold

Draw $U\sim\text{Unif}(0,1)$ and then an independent sequence $X_1,X_2,\dots$ of $\text{Unif}(0,1)$ variables. Let $N$ be the index of the first $X_n$ that is larger than $U$. Find $\mathbb{E}[N]$, or show that it is infinite.

It is infinite. Given $U=u$, $N$ is geometric with success probability $1-u$, so $\mathbb{E}[N\mid U]=\tfrac1{1-U}$. Averaging, $\mathbb{E}[N]=\int_0^1\tfrac{du}{1-u}$, which diverges logarithmically near $u=1$.