As Srikant suggests, you need to look at order statistics.
To add to Srikant's answer, you can simulate this process easily in R:
n = 10
N = 1000;sims = numeric(N)
for(i in 1:N)
sims[i] = min(runif(n))
x = seq(0,1,0.01)
lines(x, dbeta(x, 1, n), col=2)
This question is related to one of my favourite statistics problems, the German tank problem. This problem is about the maximum of uniform distributions, and can be summarised as:
Suppose one is an Allied intelligence
analyst during World War II, and one
has some serial numbers of captured
German tanks. Further, assume that the
tanks are numbered sequentially from 1
to N. How does one estimate the total
number of tanks?
Taken from wikipedia
Check out the wikipedia page for more details.