the number $x$ was drawn x from with uniform distribution of the integers {1,2,...,n} (n is unknown), what is n most likely to be? the number $x$ was drawn from with uniform distribution of the integers ${1,2,...,n}$  ($n$ is unknown), what is n most likely to be?
intuition tells me its $x$, if that's true the probability of it being chosen would be $\frac1n = \frac1x$, any other possible value would be higher then $x$ would result in a smaller probability that $x$ was be chosen.
is that a valid argument?
 A: This is called the German Tank Problem. In the case when the sample size is $1$ and the datum is $n$, the best estimate is $2n-1.$ 
A: It seems to me in order to put this question on the most solid ground you need a prior distribution on $n$ to make this work, and ideally a proper prior distribution (i.e. a real-valued distribution over the integers whose sum is 1). Just doing maximum likelihood to determine $n$ is equivalent to using an improper prior, specifically uniform over the set of positive integers (so the probability of each integer is identical and basically zero but somehow the probabilities all add up to 1, which is impossible but we pretend it's true). A proper prior distribution is much more satisfying than this. For example, with a proper prior you can come up with a confidence interval for $n$, but this is impossible with the improper "uniform" prior I described. If your prior distribution probability values are decreasing in $n$, then you will still get the result that the most likely value of $n$ is the value that you sampled. The difference is that now the result will be much more well-founded.
