Waiting to beat a random Beta threshold

Let $T, X_1, X_2, \dots$ be i.i.d. $\text{Beta}(12,8)$ random variables, and let $N$ be the first index $n\ge1$ with $X_n>T$. What is $\mathbb{E}[N]$? Answer $-1$ if it is infinite.

$-1$ (the mean is infinite). Given $T=t$, $N$ is geometric with success probability $1-F(t)$, so $\mathbb{E}[N]=\mathbb{E}\left[\tfrac{1}{1-F(T)}\right]$. Since $F(T)$ is uniform on $(0,1)$ for any continuous distribution, this is $\int_0^1\tfrac{du}{1-u}=\infty$.