Limit for the maximum of exponentials below log n

Let $X_1,X_2,\dots$ be independent $\text{Exp}(1)$ variables and $M_n=\max(X_1,\dots,X_n)$. The limit of $\mathbb{P}[M_n\le\ln n]$ as $n\to\infty$ has the form $\tfrac{b}{e}$. What is $b$?

$b=1$. The maximum is at most $\ln n$ exactly when every $X_i$ is, so the probability is $(1-e^{-\ln n})^n=(1-\tfrac1n)^n\to e^{-1}$.