Largest set allowing one-step unstructured quantum search What is the largest set admitting a deterministic quantum search algorithm, for a single marked element, that operates with only a single call to the oracle?

The question is interesting since Grover's algorithm, which for unstructured search on an $N$-element set requires $O(\sqrt{N})$ calls to the oracle, can in fact search a 4-element set using only a single call.

In general, it is interesting to ask for the minimum number of calls to a quantum oracle required to deterministically search an unstructured set of size $N$ for a single marked element.

Note that Grover's algorithm is optimal up to a constant factor in the limit of large $N$, although of course that does not mean it is optimal for any given finite set.
Perhaps a more appropriate for your question: Dong Pyo Chi and Jinsoo Kim showed that for any "Grover-like" algorithm, in which we may change the phase of the diffusion operator and the oracle gate from $-1$ to possibly independent and arbitrary complex phases, a marked element can be found with a single query if and only if there are at least $N/4$ marked items. Here is a link to their article.

Note that the case $t=N/4$ was discovered earlier by Brassard, Boyer, Hoyer and Tapp.