Here is my first try at a question, which is a really easy to state question about displaceability:

Let $D$ be the unit disk in the complex plane $D = \{ |z| \leq 1 \}$ equipped with its standard symplectic form and for $r \in (0,1)$ let $S(r) \subset D$ be the Lagrangian circle of radius $r$ centered at $0$, enclosing area $\pi r^2$.

Question: for which $r_1,\ldots, r_n$ is the Lagrangian torus $S(r_1) \times \ldots \times S(r_n)$ Hamiltonian displaceable in $D^n$?

Conjecture: for a given $r_1,\ldots, r_n$, the Lagrangian is non-displaceable iff $r_j^2 \ge 1/2$ for all $j$.

Known: Using McDuff's probes, one can see that if some $r_j^2 < 1/2$, then the Lagrangian is displaceable. I think I can show that if each $r_j^2$ lies in the set $\{ 1/2, 2/3,3/4, \ldots \}$ then the Lagrangian is non-displaceable, by embedding D^n in a product of weighted projective lines and showing the Floer homology of the Lagrangian is non-vanishing. But there must be a way of doing better.

Sub-question: (which came out of a discussion with Abouzaid): is there a Floer-theoretic way of seeing the non-displaceability of $S(r)$ for $r^2 \ge 1/2$? This is easy to prove by elementary means.