I want to see that there is no universal constant $C$ such that for every non-negative martingale $(X_n)_{n \geq 0}$, we have that $\mathbb{E}[\text{max}_{k \leq n}X_k] \leq C\mathbb{E}[X_n]$.
I've tried to do a few things with constructing a sufficiently fastly increasing submartingale but nothing has really panned out so far. I would appreciate some help.
Expanding the example mentioned in comments: let $X_0=1$, $X_{n+1}=2X_n$ with probability $1/2$ and $X_{n+1}=0$ with probability $1/2$. Then
$$\mathsf{E}\left(X_{n+1}\mid X_1,\ldots,X_n\right)=X_n$$
so $(X_n)$ is a martingale. In particular, $\mathsf{E} X_n=\mathsf{E} X_0=1$. On the other hand
$$\mathsf{E}\left(\max_{0\le k\le n} X_k\right)=\sum_{t=0}^{n-1} 2^t (2^{-t}-2^{-(t+1)})+1=\frac{n}{2}+1$$