In discrete-time, it is possible to construct a martingale which is in $L^p$ but not $L^q$ for $p<q$. To see this, simply let $(Omega, mathcal{F}, mathcal{F}_n, mathbb{P})$ be a probability space for which $(epsilon_n)$ is an iid sequence of centred random variables in $L^p setminus L^q$. Now construct the random walk $X_0=0$,

$$X_n = sum_{k=1}^n epsilon_n$$

This is now a martingale with the desired properties.

Is this possible to construct a continuous martingale with the same property? Namely, can one have a (true) continuous martingale $X_t$ such that $X_t$ has $p$-th moments, but all higher moments are infinite?