Examples of a continuous martingale with $E[\sup\limits_{0\leq s\leq t} |M_s|]=\infty$? A local martingale is a martingale iff it is in the class DL.
The condition: for every $t\in[0,\infty)$
$$E[\sup\limits_{0\leq s\leq t} |M_s|]<\infty\tag1$$
guarantees a local martingale $M$ is a martingale by ensuring it satisfies the condition for being in the class DL. Moreover, by Burkholder-Davis-Gundy, this means: for every $t\in[0,\infty)$,
$$E[\langle M\rangle^{1/2}_t]<\infty$$
My question is: do there exist (simple?) examples of continuous martingales where these two conditions are violated? 
What I am really asking is that these are known to be sufficient conditions to guarantee local martingales are martingales, why are they not necessary?
 A: There exist indeed a uniformly integrable martingale $X$ whose max $\sup_{k\in \mathbb{N}} |X_k|$ is not integrable. 
A simple example in discrete time can be found here http://www.math.fsu.edu/~nichols/martingalezoo.pdf
(see the last 2 lines of point 5). 
A related construction is found in example 4.1 of the paper by A.S. Cherny "Some particular problems of martingale theory", which produces a uniformly integrable martingale $(X_n)_{n\in \mathbb{N}}$ and a bounded process $(H_n)_{n\in \mathbb{N}}$ such that the stochastic integral $M_n:=\sum_{k=0}^n H_k (X_{k+1}-X_k)$ is a martingale which is not  uniformly integrable (by the Burkholder-Davis-Gundy inequality this implies that $\sup_{k\in \mathbb{N}} |X_k|$ is not integrable) 
A second (completely different) example in continuous time is sketched in Exercise 3.15 of the 2nd Chapter of Revuz-Yor's Book "Continuous Martingales and Brownian Motion, 3rd ed"
A: I guess the following is a counter-example: take $B$ a standard two-dimensional Brownian motion and
$$
M_t = \log ( \Vert B_t \Vert )
$$
$M$ is a local martingale (compute the SDE satisfied by $M$, rembering that $\Vert B \Vert$ is a Bessel process) , but it does not hold that
$$
E(  \langle M \rangle_t ) < \infty
$$
nor that
$$
E( \sup_{0 \le s \le t} |M_s|) < \infty.
$$
(to be checked; I am not 100% sure about this last statement).
