Fix $L \in \mathbb R^{>0}$, let $x_1, x_2, x_3, \ldots$ be IID uniformly random numbers in $[0, 1]$, and $$\mathcal N := \min \{{N \in \{1, 2, 3 , \ldots\}} \mid x_1 + x_2 + \cdots + x_N \ge L\}.$$ It's a well-known exercise that $E(\mathcal N \mid L = 1) = e$. It's not that much harder to describe the distribution for $L \le 1$ and see that $$E\left(\mathcal N \mid 0 \le L \le 1\right) = e^L.$$ Is there a good reference that describes the distribution on $\mathcal N$ for $L > 1$, where it gets more complicated?
The reference provided here, Uspensky 1937, p. 278, doesn't appear to actually discuss the distribution of $\mathcal N$.
For $n > L$, $$ \mathbb P(\mathcal N>n)=\mathbb P(x_1+\ldots+x_n <L) = \frac{1}{n!}\sum_{k=0}^{\lfloor L\rfloor} (-1)^k \binom{n}{k}(L-k)^n. $$ The r.h.s. is the CDF of Irwin–Hall distribution.
For $n\leq L$, $\mathbb P(\mathcal N>n)=1$.
To find pmf of $\mathcal N$, get $\mathbb P(\mathcal N=n)=\mathbb P(\mathcal N>n-1)-\mathbb P(\mathcal N>n)$.