This is a question asked by a student in my lecture. After drawing pictures for awhile, I thought it was a good one. I seek a nontrivial example of a pointed Heegaard diagram $(\Sigma,\mathbf{\alpha},\mathbf{\beta},z)$ of a closed, oriented 3-manifold that fails to be weakly admissible in the sense of Heegaard Floer homology.

What I call the trivial example is the diagram of $S^1\times S^2$ consisting of two parallel circles in the torus, with $z$ appropriately chosen so that there is a positive domain given by an annulus, bounded by these circles, whose index is 0. However, there are no generators of $\widehat{CF}$ in this diagram. For all I know, any Heegaard diagram that has generators is weakly admissible!

Lipshitz's index formula (Lemma 4.11 of http://arxiv.org/abs/math/0502404) seems to imply that a domain in $\widehat\pi_2(\mathbf{x},\mathbf{x})$ that violates weak admissibility either needs lots of acute corners (assuming acute corners are even available to periodic domains, it seems their negative contribution to the index should be more than canceled by $2n_{\mathbf{x}}$), or high genus (with similar cancellation coming from any entries of $\mathbf{x}$ in its interior).