Given a polyomino $P$, we can ask if it is possible for disjoint copies of $P$ to surround a single cell in the square grid - i.e., for the complement of their union to have a connected component of size $1$.
We can further refine this into polyominoes that weakly surround a cell (just covering the four edge-adjacent cells) and those that strongly surround a cell (covering all $8$ squares which share a vertex with the hole). Thanks to Julian Rosen for clarifying this distinction.
My original intuition was that this is always possible, but I was having trouble proving it; after enough struggling to show it was true, I started searching for counterexamples. Here is a polyomino with $48$ cells which does not even weakly surround a one-celled region, which is not too difficult to verify by hand:
After some modifications, I've reduced this down to a simply-connected solution with $26$ cells:
I have a size-$23$ example of a polyomino which does not strongly surround a hole:
What is the smallest polyomino that cannot surround a single-celled hole? I am interested in this question for both the weak and strong cases.
I've written some code to explore this, and have confirmed that all of the $1,227,708$ free polyominoes on at most $14$ cells can strongly surround a hole. How much can we tighten these bounds?