The Dunford-Pettis theorem states that a family $\mathcal{F}\subset L^1$ is relatively weakly compact if and only if $\mathcal{F}$ is bounded in norm and uniformly integrable, i.e. $\sup_{f\in\mathcal{F}}\lVert f \rVert_1<\infty$ and $\forall \varepsilon>0$ there exists some $\delta>0$ such that
$$ \int_A \vert f(x)\vert dx<\varepsilon $$
for every $f\in\mathcal{F}$ whenever $\vert A\vert \leq \delta$.
In the other hand, the de la Vallée-Poussin theorem says that $\mathcal{F}$ is uniformly integrable if and only if there exists a non-negative increasing convex function $\phi(t)$ such that
$$ \lim_{t\rightarrow \infty} \frac{\phi(t)}{t}=\infty \qquad \text{and} \qquad \sup_{f\in\mathcal{F}} \int \phi(\vert f(x)\vert) dx<\infty. $$
See as reference the Uniform Integrability Wiki.
With this theorems we can get a lot weakly compact set in $L^1$. For example, when $\phi(t)=t^p$ with $1<p<\infty$, we get that every set $\mathcal{F}$ bounded in the $L^p$-norm is weakly compact in $L^1$, such as the Rademacher sequence or the unit-ball of $L^2$.
So here is my question: Is there any example of a weakly compact sequence $(f_n)$ (or set $\mathcal{F}$) that is not norm-bounded in any $L^p$ space ($1<p<\infty$)? Where there can be a reference for such example? I suppose that there must be some (in Orlicz spaces and such), but I am not finding anyone.