Let's say I have balls in an urn which are labeled with the numbers $1 \cdots N$. I take a ball out of the urn, write down it's number and put it back. I will repeat this a total of $n$ times. Find the maximum likelihood estimator of $N$.
The distribution of my model is the multinomial distribution ${\mathcal{M}}_{N}(\vec{k}) =\binom{N}{\vec{k}}\prod_{i=1}^N{(\frac{1}{N})^{k_i}} $
$k_i$ is the number of times the number $i$ has been seen in the sample.
I am new to statistics and would appreciate any help in how to properly take on this question.
As I said in the comment, you don't use the number information. Firstly, think in one extraction: $$\mathbf{P}(X=x)=\frac{1}{N}\mathbf{1}\{1\leq x\leq N\}$$ where $\mathbf{1}\{\}$ is the indicator function (its value is 1 if its argument is true and its value is 0 if its argument is false). Then you repeat the experiment $n$ times, so the likelihood is: $$\mathcal{L}(N|\mathbf{x})=\left(\frac{1}{N}\right)^n\prod_{i=1}^n\mathbf{1}\{x_i\leq N\}=\left(\frac{1}{N}\right)^n\mathbf{1}\{\max(\mathbf{x})\leq N\}$$ where $\mathbf{x}=(x_1,\cdots,x_n)$ and we asume that every $x_i\in\mathbb{N}$. The last equality is proved using indicator properties: the product of indicators is 1 iff every indicator is 1 and every $x_i\leq N$ iff $\max(\mathbf{x})\leq N$. Finally, as $\left(\frac{1}{N}\right)^n$ decreasing with $N$, the maximum likelihood estimator is the minimum $N$ where the indicator function is 1. That is $\hat{N}=\max(\mathbf{x})$.