For $\theta>0$, let $X_1, X_2,\ldots,X_n$ be an iid sequence of Uniform$(0, \theta)$ random variables and we set $X_{(n)}=\max\left(X_1, X_2, \ldots, X_n \right)$.
I have already shown that the density function of $X_{(n)}$ is $$f_{X_{(n)}}(x)=\frac{n}{\theta^n}x^{n-1} \ \ \ \ \ \ 0<x<\theta$$
I have also shown (via integration by parts) that for $n=2$, the MGF of $X_{(2)}$ is $$m_{X_{(2)}}(u)=\frac{2}{\theta^2 u}\left(\theta e^{u\theta}-\frac{1}{u} e^{u\theta}+\frac{1}{u}\right)$$
I now wish to comment on the existence of the MGF of $X_{(2)}$
Clearly $u\neq 0$. To be more precise, I think that the MGF of $X_{(2)}$ exists iff $u>0$. But I am not confident in making this claim.
$0 \leq X_i \leq \theta $ for all $i$ so $0 \leq X_{(n)} \leq \theta $. Hence $ 0 \leq Ee^{tX} \leq e^{t\theta }$ for all $t \geq 0$ inlcuding $t=0$. For $t<0$ we have $Ee^{tX_{(n)}} \leq 1$. So MMG function exists for all $t \in \mathbb R$.