Expected value of smallest value Let's say we have a discrete random variable $X$ uniform distributes on $[1, n]$.  Not we make $n$ experiments to get $X$ and record the minimal value of $X$ as $X_\min$.  My question is, what is the expected value of $X_\min$?

For clarity, I prefer to segregate the sample size from the support of the distribution. That is, I will assume that $X$ is uniform over $\{ 1, 2, \ldots, U \}$. The special case that the question is concerned with is not much simpler anyway. $\newcommand{\e}{\mathrm{e}} \newcommand{\Xmin}{X_\min}$
For $1 \leqslant r \leqslant U$, the probability that the minimum $\Xmin$ is at least $r$ is equal to 
$$
\Pr(\Xmin \geqslant r) = \frac{(U-r+1)^n}{U^n}.
$$
This is because the event $\Xmin \geqslant r$ happens iff $X_i \geqslant r$ for all $i \in \{ 1, 2, \ldots, n \}$. For any fixed $i$, the probability of the latter event occurring is $\frac{U-(r-1)}{U}$. And as usual, by independence, the probability of all the $n$ events occurring simultaneously is just the product of the individual probabilities. 
Therefore (e.g., see this wikipedia page for the formula used),
$$
\mathbf{E}(\Xmin) = \sum_{r \geqslant 1} \Pr(\Xmin \geqslant r) = \sum_{1 \leqslant r \leqslant U} \frac{(U-r+1)^n}{U^n} = \frac{1}{U^n} \sum_{s = 1}^{U} s^n, \tag{$\ast$}
$$
by re-indexing the sum. 
The final expression $(\ast)$ is essentially the sum of the $n^{th}$ powers of the first $U$ natural numbers. I don't think this be simplified much further. :-)

Some asymptotics. We can say a bit more about the original question assuming $n \to \infty$. 
In this case, for constant $r$, the probability that $\Xmin \geqslant r$ is equal to
$$
\Big( 1 + \frac{1-r}{n} \Big)^n \to \e^{1-r}.
$$
Of course, if $r$ is not a constant but grows with $n$ (i.e., $r \to \infty$ as $n \to \infty$), then this probability goes to $0$. Thus the probability that the minimum is exactly $r$ is 
$$
\e^{1-r} - \e^{-r} = (\e-1) \cdot \e^{-r},
$$
which is a geometric distribution (starting at $1$). The expected value of the distribution approaches
$$
\sum_{r \geqslant 1} \e^{1-r} = \frac{1}{1 - \frac{1}{\e}} = \frac{\e}{\e - 1}.
$$
